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Every subsequence converges to L implies a_n -> L

  • #1
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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.
 

Answers and Replies

  • #2
Math_QED
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Hint: This is easy considering ##\liminf_{n \to \infty}a_n## and ## \limsup_{n \to \infty} a_n##
 
  • #3
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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##?
 
  • #4
Math_QED
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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!
 

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