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?

@BrianMScott this is not a duplicate, I am talking about a particular way of doing this problem... — TanMath, Jul 08 '15 at 02:46

score 5 · Answer 1 · answered Jul 08 '15 at 02:30

5

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

answered Jul 08 '15 at 02:30

Spenser

20,135

So my reverse BMCT won't work? – TanMath Jul 08 '15 at 02:45
and what is M and how am I supposed to use M? – TanMath Jul 08 '15 at 02:54

Prove that $\{a_n\}$ is bounded

1 Answers1