# Thread: Proving convergent seq. is a cauchy seq.

1. ## Proving 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?

2. 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 $N\leqslant m,n\implies d(x_n,x_m)\leqslant d(x_n,x)+d(x_m,x)=\varepsilon$