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!

Homework Help: Analysis (convergent sequences) help!

  1. Mar 14, 2007 #1
    I'm having a bit of trouble with two analysis questions, they are:
    1) a_n -> a iff every subsequence of {a_n} converges to a

    2) a_n->a iff {a_n} is bounded, and a is its only cluster point.

    For the first, I was thinking of doing something along the lines of saying that a subsequence of a_n would be a_(f(n)); that is, f being a function that maps the natural numbers onto itself, and has the properly that whenever n>= m, f(n) >= f(m) (nondecreasing) and from there, saying that |a_n - a| < eps given some eps(ilon), for all n> N (Where N is a natural number)... and then arguing that, since f(n) spits out a natural number, then for some f(n) >= N, a_f(n) thus has the limit....

    For the second (and probably the first too, but I can't see it?), I'm supposed to make use of the theorem:
    i) x is a cluster point of {x_n} <---> for all eps>0, N natural number, there exists an n > N such that |x_n - x| < eps, x real number
    ii) x is a cluster point of {x_n} iff there exists a converging subsequence {x_(n_k)} of {x_n} that converges to x.

    I'm unsure as to how to approach the second one.. but my idea is..
    First, work my way from a_n->a to a_n is bounded, and a is a cluster point, and work backward from there.
    However, I'm unsure as to how cluster points and limits are related...
    I'd really appreciate it if someone can point out where I could begin, and possibly clarify some things I'm confused about... if possible. :D
    Thanks for any replies!
    Last edited: Mar 14, 2007
  2. jcsd
  3. Mar 14, 2007 #2


    User Avatar
    Science Advisor
    Homework Helper

    I think you know everything you need to prove your questions but you are just sort of muddling everything together. To prove a iff b, you need to prove two things, a implies b and b implies a. So clearly state each separate implication and get started. For example 2) becomes:

    a) a_n->a implies a_n bounded.
    b) a_n->a implies a is the only cluster point.
    c) a_n bounded and a is the only cluster point of a_n implies a_n->a.

    Treat each one separately and remember proof by contradiction can be useful.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook