Every subsequence converges to L implies a_n -> L

[Mr Davis 97]

Homework Statement

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

Homework Equations

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 $\limsup_{n \to \infty} a_n$

 

[Mr Davis 97]

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!