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Cauchy sequences are bounded

Every Cauchy sequence in a metric space is bounded.

Proposition. Every in a metric space is bounded.

Proof sketch

Take ε=1\varepsilon=1 in the Cauchy condition to get NN with d(xn,xN)<1d(x_n,x_N)<1 for all n≥Nn\ge N, so the tail lies in B(xN;1)B(x_N;1). The finitely many initial terms are bounded, so all terms lie in some closed ball B′(xN;r)B'(x_N;r) for r=max⁡{d(x1,xN),…,d(xN−1,xN),1}r=\max\{d(x_1,x_N),\dots,d(x_{N-1},x_N),1\}.