So I think this is the last problem I have and I'm not thinking I'm doing it properly.
Let $a,b$ be real numbers and suppose for all $\varepsilon \gt 0, a \le b+\varepsilon$. Show that $a \le b$.
Assume $b\lt a$. Then $b+\varepsilon \lt a + \varepsilon$ and $$a \le b+\varepsilon \lt a+\varepsilon$$ If I can show that there is a contradiction I can prove it, but I'm having trouble with the contradiction.