5.5 Prove inf S≤ sup S for every nonempty subset of R. Compare Exercise 4.6(a). 4.6 Let S be a nonempty bounded subset of R. (a) Prove inf S sup S. Hint: This is almost obvious; your proof should be short.
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For every nonempty subset S of real numbers, it holds true that inf S ≤ sup S.
The infimum (inf) of a nonempty subset S of real numbers represents the greatest lower bound of S, while the supremum (sup) represents the least upper bound of S. To prove that inf S ≤ sup S for every nonempty subset S of R, we can consider the following reasoning.
Since S is a nonempty set, it contains at least one element. Let's assume that x is an element of S. As S is bounded, there exist a lower bound and an upper bound for S. Let's denote the lower bound as L and the upper bound as U.
Now, since x is an element of S, it follows that L ≤ x ≤ U. This means that x lies between the lower bound L and the upper bound U.
Therefore, L is a lower bound for S, and since the infimum is defined as the greatest lower bound, we have inf S ≤ L. Similarly, U is an upper bound for S, and since the supremum is defined as the least upper bound, we have U ≤ sup S.
Combining these inequalities, we get inf S ≤ L ≤ x ≤ U ≤ sup S. Hence, inf S ≤ sup S for every nonempty subset S of R.
Comparing this result to Exercise 4.6(a), we can see that Exercise 5.5 is a more general statement. Exercise 4.6(a) specifically deals with nonempty bounded subsets of R, whereas Exercise 5.5 holds true for any nonempty subset of R.
The concept of infimum and supremum is fundamental in real analysis. The infimum of a set represents its greatest lower bound, while the supremum represents its least upper bound. Proving that inf S ≤ sup S for every nonempty subset S of R is based on the properties of infimum and supremum.
To prove this inequality, we start by assuming that S is a nonempty subset of R. Since S is nonempty, it contains at least one element. By considering an element x in S, we can establish that the lower bound L is less than or equal to x, and x is less than or equal to the upper bound U.
Using the definitions of infimum and supremum, we conclude that inf S is less than or equal to L and L is less than or equal to x. Similarly, U is less than or equal to sup S. Combining these inequalities, we obtain inf S ≤ L ≤ x ≤ U ≤ sup S.
Thus, it is evident that inf S ≤ sup S for any nonempty subset S of R. This result highlights the relationship between the greatest lower bound and the least upper bound of a set.
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Related Questions
please help me, thanks!
Answers
Answer:
m = -2
Step-by-step explanation:
First get all the m's on one side:
5 + 2m = 3 + m
2m - m = 3 - 5 (now simplify)
m = -2
Hope this helps!
Evaluate the distance between the following pair of integers -7 and 41
Answers
Answer:
\( 41 - ( - 7) = 41 + 7 = 48\)
the event containing the outcomes belonging to a or b or both is the __________ of a and b.
Answers
The event containing the outcomes belonging to a or b or both is the union of a and b.
Let us consider two sets a and b, and say events or the elements in set a are: (x, y, z), and the elements in set b are: (o, p, q)
The union of these two sets will give:
aUb=(x, y, z, o, p, q)
Now, a union of two sets (denoted by U), for our case a and b, is the set or event containing the elements belonging to a or b or both the elements in a and b altogether.
Thus, the even containing the outcomes belonging to a or b or both in a and b altogether is the union of a and b.
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two tennis balls fall through the air from a tall building. one of them is filled with lead pellets. the ball to reach the ground first is the a) regular ball. b) lead-filled ball. c) same for both
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If two tennis balls fall through the air from a tall building. one of them is filled with lead pellets. the ball that reaches the ground first is the lead-filled ball. Thus, option b is correct.
The lead-filled ball will hit the ground before the air-filled ball does because of its greater acceleration and larger distance covered in the same period of time.
The basic idea behind this image is that larger things strike the ground first since there is less air resistance against them. In contrast to other balls, the lead-filled ball is heavier when it is filled with air, which causes it to accelerate more under the influence of gravity.
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A fish is swimming at a constant rate toward the ocean floor. The equation y = -7x - 3 can be used to represent this situation, where y is the depth of the fish in meters below sea level and x s represents of seconds the fish has been swimming.
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C. From a starting position of 3 meters below sea level, the fish is descending 7 meters per second.
How to determine which statement best describes the depth of the fish, given the equation?The equation of a line is an algebraic form of representing the set of points, which together form a line in a coordinate system.
The general form of the equation of a line is y = mx+b,
where m is the slope (rate of change) and b is the y-intercept (starting position)
In this case, the equation y = -7x - 3 can be used to represent this situation.
Thus, the starting position is 3 meters below sea level (-3) and the fish is descending 7 meters per second (-7).
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Complete Question
A fish is swimming at a constant rate toward the ocean floor. The equation y = -7x - 3 can be used to represent this situation, where y is the depth of the fish in meters below sea level and x is the number of seconds the fish has been swimming.
Which statement best describes the depth of the fish, given this equation?
A. From a starting position of 7 meters below sea level, the fish is descending 3 meters per second.
B. From a starting position of 7 meters below sea level, the fish is ascending 3 meters per second.
C. From a starting position of 3 meters below sea level, the fish is descending 7 meters per second.
D. From a starting position of 3 meters below sea level, the fish is ascending 7 meters per second.
Plllllllllzzzzzzz I I need a little help so don’t be shy and I know it’s a lot but it’s worth it
So thanks
Answers
Answer:
The wheel will make 8 Rotations.
The Bike will travel 75 feet
Step-by-step explanation:
Wheel: 40 feet / 5 feet per rotation = 8 rotations
Bike: 5 feet per rotation * 15 rotations = 75 feet traveled
Which values for A and B will create infiniely many solutions for this system of equations
Answers
Answer :
4x-ay=15
Step-by-step explanation:
8) Let R be a relation that is reflexive and transitive. Prove that R2 = R for any R with these two properties. 9) Suppose that the relation R is anti-reflexive. Is R2 necessarily anti-reflexive? Give a reason for your answer.
Answers
Even if R is anti-reflexive, R2 may not necessarily be anti-reflexive. It depends on the specific properties and composition of the relation R.
Let R be a relation that is reflexive and transitive. We want to prove that R2 = R for any relation R with these two properties.
To prove this, we need to show that for any ordered pair (a, b), (a, b) ∈ R2 if and only if (a, b) ∈ R.
First, let's consider (a, b) ∈ R2. By definition, (a, b) ∈ R2 means that there exists an element c such that (a, c) ∈ R and (c, b) ∈ R.
Since R is reflexive, we know that (a, a) ∈ R and (b, b) ∈ R.
By the transitivity of R, if (a, c) ∈ R and (c, b) ∈ R, then (a, b) ∈ R.
Therefore, (a, b) ∈ R2 implies (a, b) ∈ R.
Now, let's consider (a, b) ∈ R. Since R is reflexive, we have (a, a) ∈ R and (b, b) ∈ R.
By the definition of R2, (a, a) ∈ R2 and (b, b) ∈ R2.
Since R is transitive, if (a, a) ∈ R2 and (a, b) ∈ R2, then (a, b) ∈ R2.
Therefore, (a, b) ∈ R implies (a, b) ∈ R2.
We have shown that for any ordered pair (a, b), (a, b) ∈ R2 if and only if (a, b) ∈ R. Hence, R2 = R.
If the relation R is anti-reflexive, it is not necessarily true that R2 is anti-reflexive.
To understand why, let's consider an example. Let R be a relation defined on the set of integers such that R contains the ordered pairs (a, b) where a < b.
In this case, R is anti-reflexive because for any integer a, (a, a) is not in R.
Now, let's consider R2. R2 is the composition of R with itself. If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R2.
In our example, if we take a = 1, b = 2, and c = 3, we have (1, 2) ∈ R and (2, 3) ∈ R. Therefore, (1, 3) ∈ R2.
However, (1, 1) is not in R2 because (1, 1) is not in R. Therefore, R2 is not anti-reflexive in this case.
This example demonstrates that even if R is anti-reflexive, R2 may not necessarily be anti-reflexive. It depends on the specific properties and composition of the relation R.
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plz help asap!!!! thank you!!!........
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Answer:
choice 1) 312 cm²
Step-by-step explanation:
area = (16 x 12) + (10 x 12) = 312 cm²
Answer:
312
Step-by-step explanation:
triangle：
10x12/2=60
two triangle：
60x2=120
rectangle：
16x12=192
add up：
120+192=312
I GIVE BRAiNIEsT THInG S HeLP iM TImED
Answers
Answer:
4
Find the value of given expression when p = -2 and q = -1
( ( p² ) ( q ^ -3 ) ) ^ -2 × ( ( p ^ -3 ) (q⁵ ) )^ -2= ( 4 × ( -1) )^ -3 × ( -1 / 8 × -1 ) ^ -2
= 1 / 16 × 64
= 64/16
= 4
Hope this answer will help you.
Prove OPR=QRP using a paragraph proof. (Write out the words for symbols. For example, write "Z" as "angle" and "x" as
"congruent to.")
Answers
Answer:
\(\triangle OPR \cong \triangle QRP\) as per AAS congruence.
Step-by-step explanation:
We are given two triangles \(\triangle OPR \ and\ \triangle QRP\).
Side OP is parallel to Side QR.
OP || QR
\(\angle O = \angle Q\)
To prove:
The two triangles are congruent i.e. \(\triangle OPR \cong \triangle QRP\)
Solution:
Let us have a look at the triangles:
\(\triangle OPR \ and\ \triangle QRP\)
1. Given that \(\angle O = \angle Q\).
2. Side OP || RQ so, \(\angle OPR = \angle QRP\) because they are the alternate angles of parallel sides. (Alternate angels are equal when a line cuts two parallel sides).
3. Side PR is common to both the sides i.e. PR = RP
Hence, by AAS congruence i.e. two angles and a side which is not between the two angles are same.
\(\therefore \triangle OPR \cong \triangle QRP\) as per AAS congruence.
HELP!!!
How many millimeters long is the missing side?
There is a screenshot of it below
Answers
Answer:
8
Step-by-step explanation:
18-12
12-X
X=12*12/18
X=8mm
On the Guardian lines a and B
Answers
Answer:
2 and - 1
Step-by-step explanation:
Calculate the gradients using the slope formula
m = \(\frac{y_{2}-y_{1} }{x_{2}-x_{1} }\)
with (x₁, y₁ ) = (- 2, -3) and (x₂, y₂ ) = (3, 7) ← 2 points on line A
\(m_{A}\) = \(\frac{7+3}{3+2}\) = \(\frac{10}{5}\) = 2
with (x₁, y₁ ) = (- 2, 7) and (x₂, y₂ ) = (7, - 2) ← 2 points on line B
\(m_{B}\) = \(\frac{-2-7}{7+2}\) = \(\frac{-9}{9}\) = - 1
Eight runners are competing in the 100-meter dash. In how many different ways can the runners finish
Answers
There are 40320 different ways of the runners finishing.
Since there are 8 runners, and the order in which they finish matters. This is a permutation question.
The PermutationsSo, there are 8 person for the first position, 7 persons for the second position, 6 persons for the third position, 5 persons for the fourth position, 4 persons for the fifth position, 3 persons for the sixth position, 2 persons for the seventh position and 1 person for the last position.
So, in all there are 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 8! ways of finishing.
8! = 40320 ways.
So, there are 40320 different ways of the runners finishing.
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What is 25% of 30??????
Answers
Answer:
7.5
Step-by-step explanation:
First, turn 25% into a decimal. You do this by dividing 25 by 100.
25/100 = 0.25
Now, multiply 0.25 by 30.
30 x 0.25 = 7.5
if you have 2, 6 sided cubes and you can add any numbers to them, which numbers do you put on them so that you can display every day of the month?
Answers
Hope this helped ; )
Answer: its 12 and 8 10 12 14 16 either of those
Step-by-step explanation:
suppose you are researching the eating habits of people your age. what sampling method could you use to find the percent of students in your grade who eat five servings of fruit and vegetables each day? what is an example of a survey question that does
Answers
A random sampling method could be used to find the percentage of students in your grade who eat five servings of fruit and vegetables each day.
Why is this the best sampling method to be used?This method involves selecting a random sample of students from the entire population of your grade and surveying them about their eating habits.
In other word, its involves randomly selecting a subset of students from the entire grade population and collecting data on their eating habits. By selecting students at random, the sample is more likely to be representative of the entire grade. This cam provide accurate insights into the percentage of students meeting the desired dietary recommendation.
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The ratio of number of men to the number of women in the auditorium is 4:7. The number
of men is 487.
2.3.1 Determine the number of women in the auditorium.
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The ratio 4:7 means for every 4 men there are 7 women.
Divide total number of men by 4:
487/4 = 121.75
Multiply that by 7:
121.75*7 = 852.25 Round to the nearest whole number.
There is approximately 852 women.
Victor used a 36-month line of
credit for $15,000 to remodel his
kitchen. If the interest rate is 2. 5%,
how much will he pay in interest?
Answers
Answer:
Step-by-step explana2 4
457
6.46
PLS HELP ASAP THANKS ILL GIVE BRAINLKEST
Answers
Answer: D (n> 8/ 3 )
Write a question that has the quotient of -3/8
Answers
What is a concave hexagon with 2 pairs of congruent sides?
Answers
A concave hexagon with 2 pairs of congruent sides is a shape that has six sides and two pairs of sides that are of equal length, but the shape curves inward at one or more angles, creating a concave indentation.
A hexagon is a six-sided polygon, and when two pairs of its sides are congruent,
it means that four of its sides have the same length, if the shape curves inward at one or more angles,
it creates a concave hexagon, which means that the indentation on the shape is facing inward, rather than outward, a concave hexagon with 2 pairs of congruent sides is a six-sided polygon with four sides of equal length, but with a curvature that creates an inward indentation.
Hence, a concave hexagon with 2 pairs of congruent sides is a six-sided figure with an indentation and two sets of sides with equal lengths.
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the sscp exam consists of ____ multiple-choice questions, and must be completed within three hours.
Answers
Answer:
125 questions
Answer:
125 questions
Step-by-step explanation:
What do you learn at pirate school?
Answers
2.) x=11/3
3.) x=21
4.) x=-12
5.) x=19/21
6.) x=-2/5
10.) x=13/12
11.) x=5/4
answer: the three arrrs
complete the equation to show how many fifths are equavalent to 6/10
Answers
Three-fifths is equivalent to 6/10.
We must build up the equation and solve for the unknown value to determine how many fifths are equal to 6/10.
Let's use "x" to stand in for the unknown value. One way to set out the equation is as follows:
x/5 = 6/10
We can multiply both sides of the equation by 10 to remove the denominators:
10 * (x/5) = 10 * (6/10)
Simplifying:
2x = 6
adding two and dividing both sides of the equation:
2x/2 = 6/2
x = 3
3 fifths are therefore comparable to 6/10.
Thus, There are three-fifths to make 6/10.
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You pick a card at random.
2 3 4 5 6
What is P(greater than 4)?
Write your answer as a percentage.
Answers
Answer:
\(40\%\)
Step-by-step explanation:
\(P(X > 4)=P(5)+P(6)=\frac{1}{5}+\frac{1}{5}=\frac{2}{5}=40\%\)
Answer:
P(greater than 4) = \(40\%\)
Step-by-step explanation:
There are 5 given numbers (2 , 3 , 4 , 5 , 6). You are solving for the probability of obtaining a card greater than (and not equal to 4). In this case, there will be two numbers (5 & 6). Find the probability:
P(greater than 4) = (Amount of numbers greater than 4/Total amount of numbers) = \(\frac{2}{5}\).
Next, find the percentage:
\(\frac{2}{5} * \frac{20}{20} = \frac{40}{100} = 0.40 = 40\%\)
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the average bmi for a sample of 10 preschoolers is 16.3, with a standard deviation of 1.4. what is the 90% confidence interval for the bmi of all preschoolers? (round to two decimal places) group of answer choices (15.57, 17.03) (15.29, 16.91) (15.3, 17.3) (15.49, 17.11)
Answers
The 90% confidence interval for the BMI of all preschoolers is option (d) (15.49, 17.11)
To calculate the 90% confidence interval for the BMI of all preschoolers, we can use the formula
CI = X ± Z × (σ / sqrt(n))
where X is the sample mean, σ is the sample standard deviation, n is the sample size, and Z is the z-score corresponding to the desired confidence level (in this case, 90%).
We can find the value of Z using a standard normal distribution table or calculator, which gives us a value of 1.645.
Plugging in the values from the problem, we get
CI = 16.3 ± 1.645 × (1.4 / sqrt(10))
CI = 16.3 ± 0.90
So the 90% confidence interval for the BMI of all preschoolers is (15.49, 17.11).
Therefore, the correct option is (d) (15.49, 17.11)
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Calculation: According to Fedex’s 2017 annual report, the operating revenue in 2017 was $60,319 million while its operating expenses were $55,282 million. Calculate the operating ratio of Fedex in 2017 (show your work).
Answers
The operating ratio of FedEx in 2017 was approximately 91.64%.
To calculate the operating ratio of FedEx in 2017, we need to divide the operating expenses by the operating revenue and then multiply the result by 100 to express it as a percentage.
Operating Ratio = (Operating Expenses / Operating Revenue) * 100
Given:
Operating Revenue = $60,319 million
Operating Expenses = $55,282 million
Calculating the operating ratio:
Operating Ratio = (55,282 / 60,319) * 100
Operating Ratio ≈ 91.64%
Therefore, the operating ratio of FedEx in 2017 was approximately 91.64%.
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gabriella went skiing. she paid $35 to rent skis and $15 an hour to ski. if she paid a total of $95, how many hours did she ski?
Answers
Gabriella skied for 6 hours, Let x be the number of hours that Gabriella skied. We know that she paid $35 for ski rental and $15 per hour for skiing,
for a total of $95. We can set up the following equation to represent this information:
35 + 15x = 95
Solving for x, we get:
15x = 60
x = 4
Therefore, Gabriella skied for 6 hours.
Here is a more detailed explanation of how to solve the equation:
Subtract $35 from both sides of the equation.
15x = 60
15x - 35 = 60 - 35
15x = 25
Divide both sides of the equation by 15.
15x = 25
x = 25 / 15
x = 4
Therefore, x is equal to 4, which is the number of hours that Gabriella skied.
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Module 13 area and polygons answer key please help.
Answers
Answer:
Parallelogram - B.
Trapezoid - C.
Rhombus - A.
Step-by-step explanation: