# Every Cauchy net is convergent [duplicate]

Prove that in a Banach space every Cauchy net is convergent.

Edit:Let $A$ be a directed set and $\{f_{\alpha}\}_{\alpha\in A}$ is a net in $X$ topological space then $\{f_{\alpha}\}_{\alpha\in A}$ is cauchy net if $\forall \varepsilon>0$ there is $\alpha_0$ such that $$\|f_{\alpha_1}-f_{\alpha_2}\|<\varepsilon \hspace{0.5cm} \forall \alpha_1,\alpha_2\geq\alpha_0$$
• @user161825 thanks.But i don't understand that why "sequence $\langle x_{i(k)}:k\in\Bbb N\rangle$ is $d$-Cauchy" – Rizwan Ahmed Aug 13 '14 at 8:58
• @RizwanAhmed. Pick $\varepsilon>0$. Choose $N\in\mathbb N$ so large that $2^{-N}<\varepsilon$. If $n,m$ are natural numbers exceeding $N$ and $n\geq m$ (with no loss of generality), then $$d(x_{i(n)},x_{i(m)})\leq2^{-m}<2^{-N}<\varepsilon$$ by the construction of the sequence $\{x_{i(k)}\}_{k\in\mathbb N}$ from the Cauchy net $\{x_i\}_{i\in I}$. – triple_sec Aug 13 '14 at 9:03