# Every subsequence converges to L implies a_n -> L

## Homework Statement

Let ##\{a_n\}## be a bounded sequence such that every convergent subsequence has limit ##L##. Prove that ##\lim_{n\to\infty}a_n = L##.

## The Attempt at a Solution

I'm not really understanding this problem. Isn't ##\{a_n\}## a subsequence of itself? So isn't it immediately the case that ##\lim_{n\to\infty}a_n = L## by the hypothesis?

EDIT: Nevermind. I notice now that it says every convergent subsequence.

member 587159
Hint: This is easy considering ##\liminf_{n \to \infty}a_n## and ## \limsup_{n \to \infty} a_n##

Hint: This is easy considering ##\liminf_{n \to \infty}a_n## and ## \limsup_{n \to \infty} a_n##
Could we just say that since the set of subsequential limits is just ##\{L\}##, we have that ##\limsup a_n = \sup \{L\} = L = \inf \{L\} = \liminf a_n##. So ##\lim_{n\to\infty} a_n = L##?

member 587159
Could we just say that since the set of subsequential limits is just ##\{L\}##, we have that ##\limsup a_n = \sup \{L\} = L = \inf \{L\} = \liminf a_n##. So ##\lim_{n\to\infty} a_n = L##?

Exactly. Well done!