Cauchy Sequences Converge

Every convergent sequence is a Cauchy sequence. (

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Proof

Suppose $(x_{n})$ converges to $x$ . To prove $(x_{n})$ is a Cauchy Sequence we must find a point in the sequence after which we have $| x_{n} - x_{m} | < ϵ$ . This can be done using an application of the triangle inequality as follows if we choose $m \geq n \geq N$ :

\begin{aligned} | x_{n} - x_{m} | & = | x_{n} - a + (a - x_{m}) | \\ \leq \underset{< \frac{ϵ}{2}}{\underset{⏟}{| x_{n} - a |}} + \underset{< \frac{ϵ}{2}}{\underset{⏟}{| x_{m} - a |}} \\ < ϵ \end{aligned}

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