Prove: If $\lim_{n\to\infty}a_n = L$ and $a_n > a$ for all $n$ then $L \geq a$
Proof: We know from the definition of the limit that $\forall_{\epsilon > 0} \exists_n s.t. \forall_{n>N} |a_n - L| < \epsilon$. Now since $a_n > a$ for all $n$...
I am not really sure where to go from here. Is it the case that all sequences defined by this statement are monotone non-increasing? Then intuitively we could say $a_n = L$ for sufficiently large $n$. Thus, by transitivity $L > a$