Theorem:Bolzano-Weierstrass

Theorem (Bolzano—Weierstrass)

Every bounded sequence of real numbers contains a convergent subsequence.

Let (x_n) be a sequence of real numbers bounded by a real number M, that is |x_n| < M for all n. We define the set A by A = {r | |r| ≤ M and r < x_n for infinitely many n}. We note that A is non-empty since it contains −M and A is bounded above by M. Let x = sup A.

We claim that, for any ε > 0, there must be infinitely many points of x_n in the interval (x − ε, x + ε). Suppose not and fix an ε > 0 so that there are only finitely many values of x_n in the interval (x − ε, x + ε). Either x ≤ x_n for infinitely many n or x ≤ x_n for at most only finitely many n (possibly no n at all). Suppose x< x_n for infinitely many n. Clearly in this case x ≠ M. If necessary restrict ε so that x + ε ≤ M. Set r = x + ε/2 we have that r < x_n for infinitely many n because there are only finitely many x_n in the set [x,r] and x must be less than infinitely many x_n, furthermore |r| < M. Thus r is in A, which contradicts that x is an upper bound for A. Now suppose x< x_n for at most finitely many n. Set y = x − ε/2. Then there are at most only finitely man n so that x_n ≥ y. Thus, if r < x_n for infinitely many n, we have that r ≤ y. This means that y is an upper bound for A that is less than x, contradicting that x wast the least upper bound of A. In either case we arrive at a contradiction, thus we must have that for any ε > 0, there must be infinitely many points of x_n in the interval (x − ε, x + ε).

Now we show there is a subsequence that converges to x. We define the subsequence inductively, choose any x_n1 from the interval (x − 1, x + 1). Assuming we have chosen x_n1, …, x_nk−1, choose x_nk to be an element in the interval (x − 1/k, x + 1/k) so that n_k∉{n₁, …, n_k−1}, this is possible as there are infinitely many elements of (x_n) in the interval. Notice that for this choice of x_nk we have that |x − x_nk|<1/k. Hence for any ε>0, if we take any k > 1/ε, then |x_nk-x| < ε. That is the subsequence (x_nk) → x. $\Box$

Resources

• Sequences, Wikibooks: Real Analysis