Proving convergent seq. is a cauchy seq.

**summerset353** · February 17th 2010, 05:41 PM

I proved convergent sequences and cauchy sequences.
How would I prove that every convergent sequence is a Cauchy sequence?

**Drexel28** · February 17th 2010, 07:24 PM

Originally Posted by summerset353

I proved convergent sequences and cauchy sequences.
How would I prove that every convergent sequence is a Cauchy sequence?

Let $\varepsilon>0$ be given. There exists some $N\in\mathbb{N}$ such that $N\leqslant n\implies d(x_n,x)<\frac{\varepsilon}{2}$ and so $N\leqslant m,n\implies d(x_n,x_m)\leqslant d(x_n,x)+d(x_m,x)=\varepsilon$

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