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Explanation of the Bolzano-Weierstrass theorem proof

  1. Oct 7, 2012 #1
    1. The problem statement, all variables and given/known data

    Every bounded sequence has a convergent subsequence.

    2. Relevant equations

    Suppose that closed intervals I_0 [itex]\supset[/itex] I_1[itex]\supset[/itex] ... [itex]\supset[/itex] I_m and natural numbers n[itex]_{1}[/itex] < n[itex]_{2}[/itex] <
    ... < n[itex]_{m}[/itex] have been chosen such that for each 0 [itex]\leq[/itex] k [itex]\leq[/itex] m,
    |I[itex]_{k}[/itex]| = [itex]b-a/2^{k}[/itex], x[itex]_{n}_{k}[/itex][itex]\in[/itex]I[itex]_{k}[/itex]n and x[itex]_{n}[/itex] [itex]\in[/itex] I[itex]_{k}[/itex] for infinitely many n.

    3. The attempt at a solution
  2. jcsd
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