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
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)