Bounded sequence implies convergent subsequence

Join the discussion
Ask a follow-up here, or get your own question answered by working scientists, mathematicians and engineers — people, not an autocomplete.
Real named experts · corrections over time · the nuance an AI answer skips
Sign up Log in
1 reply · 11K views
Scousergirl
Messages
15
Reaction score
0
How can you deduce that nad bounded sequence in R has a convergent subsequence?
 
Physics news on Phys.org
This is the Bolzano–Weierstrass theorem. Any decent calculus or analysis textbook should have a proof of it. Or search online. Or use this hint: Prove that a bounded sequence of real numbers has a monotone subsequence.
 

Similar threads

Proving convergence of sequence from convergent subsequences
  • lys04
  • · Replies 15 ·
  • Calculus and Beyond Homework Help
Replies
15
Prove: A bounded sequence contains a convergent subsequence.
  • Eclair_de_XII
  • · Replies 6 ·
  • Calculus and Beyond Homework Help
Replies
6
Bounded and monotonic sequences - Convergence
  • chwala
  • · Replies 7 ·
  • Calculus and Beyond Homework Help
Replies
7
Prove that the sequence does not have a convergent subsequence
  • fabiancillo
  • · Replies 4 ·
  • Calculus and Beyond Homework Help
Replies
4
Does a Bounded, Divergent Sequence Always Have Multiple Convergent Subsequences?
  • Bashyboy
  • · Replies 8 ·
  • Calculus and Beyond Homework Help
Replies
8
Every subsequence converges to L implies a_n -> L
  • Mr Davis 97
  • · Replies 3 ·
  • Calculus and Beyond Homework Help
Replies
3
Unit normed linear functional on a space of sequences
  • docnet
  • · Replies 13 ·
  • Calculus and Beyond Homework Help
Replies
13
Lemma: Extracting a convergent subsequence
  • Hall
  • · Replies 11 ·
  • Calculus and Beyond Homework Help
Replies
11
Proving convergence of rational sequence
  • member 731016
  • · Replies 2 ·
  • Calculus and Beyond Homework Help
Replies
2
Convergence of a recursive sequence
  • Mr Davis 97
  • · Replies 3 ·
  • Calculus and Beyond Homework Help
Replies
3