Cauchy Sequences Are Bounded

Bounded Cauchy Sequences

Proof

Given $ϵ = 1$ then $\exists N$ where $| x_{m} - x_{n} | < 1$ for all $m, n \geq N$ . Thus, we must have $| x_{n} | < | x_{N} | + 1$ for all $n \geq N$ . It follows:

M = max (| x_{1} |, \dots, | x_{N - 1} |, | x_{N} | + 1)

is a bound for the sequence $(x_{n})$ .
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