(a) If sup A < sup B, show that there exists an element of $b \in B$ that is an upper bound for $A$.

I have argued that if sup A $\lt$ sup B, then choose an $\epsilon>0$ such that sup A +$\epsilon \in B$. Since $a \le $ sup A for all $a \in A$, it follows $a \lt $ sup A + $\epsilon$ hence sup A + $\epsilon $ is an upper bound of A as well as an element of B.

(b) Give an example to show that this is not always the case if we only assume sup A $\le$ sup B.

I am having trouble extending my argument into an example which makes me think that it may not be correct. 

Question: Is my argument for (a) correct? If not how would one show this fact? Can you give an example asked for in (b)? Thanks in advance.

I need to understand how I am assuming what I am supposed to prove in my approach. Can someone elaborate on that please? The other question doesn't really address this issue.