Math Help - Supremum Proof

1. Supremum Proof

I need help to prove the following question -
"Let S be a set (S⊂R) and let a be a positive number. Define the set T by T={as: s ∈ S}. Prove supT=asupS" Im not sure how to go about this proof whether it would be using an epsilon style proof or otherwise any help would be appreciated

2. Hi,
let $S$ be nonempty and bounded above, then $T$ is also nonempty and bounded above, so both suprema exist. Let's denote $q = \mbox{sup }S$.
Is $aq$ an upper bound of $T$? Yes, because any $t \in T$ can be written as $t = as$ for some $s \in S$ and we have $s\leq q$ and $a$ is positive, so $t=as \leq aq$.
Is $aq$ the least upper bound? If it were not, there would be some $p \in \mathbb{R}$ such that $p and $p$ is an upper bound of $T$. Then $p/a$ is an upper bound of $S$, because for any $s\in S$ we have $as \in T$ so $as \leq p$ and $s\leq p/a$. Furthermore, $p/a < q$. But this would mean that $q$ is not the least upper bound of $S$, which is contradiction.
We conclude that $aq = \mbox{sup }T$.