Ph.D. QUALIFYING EXAM IN REAL ANALYSIS
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چکیده
Solution: Since {xn} ∞ n=1 is a bounded sequence in R, the Bolzano-Weierstrass Theorem ensures that there is at least one convergent subsequence: let L be its limit. If the entire sequence does not converge to L, then for some > 0 there must be infinitely many terms of the sequence outside the ball B = (L− , L+ ). This infinite but bounded collection of terms likewise by Bolzano-Weierstrass has a convergent subsequence; but the latter limit lies in the closed set B , hence is different from L. This contradiction shows {xn} ∞ n=1 converges to L.
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