Science:Math Exam Resources/Courses/MATH220/April 2011/Question 10 (a)

MATH220 April 2011

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Question 10 (a)
Let $A\subset \mathbb {R}$ be a bounded set of real numbers and let $B\subseteq A$ be a nonempty subset of A. Prove that B is also bounded and that $\inf(A)\leq \sup(B)\leq \sup(A).$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Recall the definition of supremum and infimum of sets of real numbers.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. By definition of the supremum of a set, sup(B) is the smallest real number that is larger or equal to all the numbers in the set B. Since all the numbers of the set B are also numbers of the set A, we can conclude that sup(A) is also larger or equal to all the numbers of the set B (since it is larger or equal to all numbers in the set A) but since sup(B) is the smallest real number with that property we can conclude that $\sup(B)\leq \sup(A)$ Now inf(A) is a number that is less or equal than any number in the set A and so since B is a subset, it is in particular less or equal to any number in the set B. Since sup(B) is larger than any number in the set B we can conclude that $\inf(A)\leq \sup(B)$ Note: we could actually prove the more interesting and stronger statement: $\inf(A)\leq \inf(B)\leq \sup(B)\leq \sup(A).$

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