How Does Every Convergent Subsequence Prove the Limit of a Sequence?

[davitykale]

Homework Statement

Prove that lim a_n = L given that (a_n) is a sequence defined such that every subsequence has a convergent that converges to L\in R

Homework Equations

Bolzano-weierstrass and the works

The Attempt at a Solution

A subsequence of a subsequence is just a subsequence of the original sequence, so if I can show that (a_n) is bounded. Can I do something with maybe 2L as a bound?

 

[davitykale]

I'd really appreciate any help I could get :)

 

[Redbelly98]

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