I proved convergent sequences and cauchy sequences.
How would I prove that every convergent sequence is a Cauchy sequence?
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 $\displaystyle N\leqslant m,n\implies d(x_n,x_m)\leqslant d(x_n,x)+d(x_m,x)=\varepsilon$