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I'm trying to prove the supremum of a set. In class my teacher went over an example and he proved that sup(−3, 4) = 4. In his proof he used the fact that the supremum had to be positive because he could pick a positive number in the set.

But in the homework assignment (Suppose that a ∈ R. Prove that sup(−∞, a) = a.) I'm pretty sure I can't assume a to be positive?

I'm not looking for someone to do the proof for me I'm just not sure how to structure my proof. If there is a technique that applies like induction or epsilon delta proofs.

This is for a first year undergrad course intro to analysis.

Any help would be very much appreciated. Thanks in advance

jack
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1 Answers1

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Hint:

1) Prove that $a$ is upper bound (for all $x\in(-\infty,a)$ you have $x\le a$)

2) Prove by contradiction that $a$ is least upper bound (if there exists an upper bound smaller than $a$, say $a-\epsilon$, then you can find an element in $x \in (-\infty,a)$ which is larger than $a-\epsilon$, e.g. $a-\epsilon/2$, contradiction)

Momo
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