I proved convergent sequences and cauchy sequences.

How would I prove that every convergent sequence is a Cauchy sequence?

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- Feb 17th 2010, 05:41 PMsummerset353Proving convergent seq. is a cauchy seq.
I proved convergent sequences and cauchy sequences.

How would I prove that every convergent sequence is a Cauchy sequence? - Feb 17th 2010, 07:24 PMDrexel28
Let $\displaystyle \varepsilon>0$ be given. There exists some $\displaystyle N\in\mathbb{N}$ such that $\displaystyle N\leqslant n\implies d(x_n,x)<\frac{\varepsilon}{2}$ and so $\displaystyle N\leqslant m,n\implies d(x_n,x_m)\leqslant d(x_n,x)+d(x_m,x)=\varepsilon$