Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Every bounded sequence is Cauchy?

  1. Nov 2, 2014 #1
    I've been very confused with this proof, because if a sequence { 1, 1, 1, 1, ...} is convergent and bounded by 1, would this be considered to be a Cauchy sequence? I'm wondering if this has an accumulation point as well, by using the Bolzanno-Weirstrauss theorem.

    I really appreciate the help guys. Thanks.
  2. jcsd
  3. Nov 2, 2014 #2
    Cauchy is the same thing as convergent in the real numbers (or any complete metric space). A sequence can be bounded but not converge, like 0, 1, 0, 1, 0, 1...

    However, the Bolzano-Weierstrass theorem says that there is a convergent subsequence, which in this case is easy to find directly: 0,0,0,0,0... or 1,1,1,1,1...

    You get those by either skipping all the 1's or skipping all the 0's. The full sequence 0, 1, 0, 1, 0, 1... therefore has the accumulation points 0 and 1, since it has subsequences that converge to those points.

    1, 1, 1, 1 has the accumulation point 1, the thing that it converges to.
  4. Nov 2, 2014 #3


    Thanks a lot, now it makes sense. So, not every bounded sequence is cauchy, but it's subsequences can be convergent (cauchy).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook