1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prove: Cauchy sequences are converging sequences

  1. Oct 15, 2012 #1
    1. The problem statement, all variables and given/known data
    I want to prove that if a sequence a[n] is cauchy then a[n] is a converging sequence

    2. Relevant equations
    What I know is:
    1. a[n] is bounded
    2. any subsequence is bounded
    3. there exists a monotone subsequence
    4. all monotone bounded sequences converge
    5. there exists a convergent subsequence by the above
  2. jcsd
  3. Oct 15, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I assume your sequence is real-valued, since you are talking about monotone subsequences.

    What you have done so far is fine, assuming you have proofs for each claim. So now you need to prove that any Cauchy sequence which has a convergent subsequence must converge.

    Let [itex]L[/itex] be the limit of the subsequence [itex](a_{n_k})[/itex]. Given [itex]\epsilon > 0[/itex], there exists a natural number [itex]N[/itex] such that [itex]|a_{n_k} - L| < \epsilon[/itex] for all [itex]n_k \geq N[/itex]. There also exists a natural number [itex]M[/itex] such that [itex]|a_n - a_m| < \epsilon[/itex] whenever [itex]n,m \geq M[/itex]. How can you combine these two pieces of information to conclude that [itex]a_n[/itex] converges to [itex]L[/itex]?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook