Problem: Suppose $a_n \to a$ and $a_n \geq b$ for all $n$. Show that $a \geq b$.
My proof: Assume that $b > a$ and let $\epsilon = b - a$. By definition if $\{a_n\}$ converges then $|a_n - a| < \epsilon$ for $n \geq N$. By expanding the absolute value, I see that $a_n - a < b- a$. By adding $a$ to both sides, we get that $a_n < b$ a contradiction.
Is my proof correct? And is there any way to tackle a problem like this without constructing an $\epsilon$?