Prove that $\{a_n\}$ is bounded

Question

Let $\{a_n\}$ be a sequence an $L$ a real number such that $\lim_{n\to\infty} a_n = L$ Prove that $\{a_n\}$ is bounded

This reminds me of the bounded monotone convergence theorem (BMCT) but in reverse. So I was thinking of proving by somehow reversing the proof of the BMCT. Am I approaching this correct? And are there alternative ways of doing this that may be simpler?

Answer 1

Take $N\in\Bbb N$ such that $|a_n-L|<1$ for $n\ge N$ and let $$M=\max\{|a_1|,\ldots,|a_N|,L+1\}.$$