Can it be shown that the limit of a bounded sequence is no greater than the bounding value? Let $(a_n)$ be a convergent sequence. Since $(a_n)$ converges it is bounded and therefore there exists a number $\alpha \geq 0$ such that $|a_n| \leq \alpha \; \forall \; n \in \mathbb{N}$. Is it true $\lim_{n \to \infty} |a_n| \leq \alpha$  ?
I believe it is true and my proposed answer to this question will attempt to confirm this belief.

As a general tip, usually when I believe something is true, a proof by contradiction is in order. In this case if $a_{n} \to a$, but $a>\alpha$ then $a-\epsilon>\alpha$ for some $\epsilon>0$. By definition of convergence, there is some $a_N \in (a-\epsilon, a+\epsilon)$, but then $a_N > a-\epsilon >\alpha$, contrary to $\alpha$ being an upper bound for $(a_n)$.
