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

I have trouble to prove this, please help.Thanks

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 $$\|f_{\alpha_1}-f_{\alpha_2}\|<\varepsilon \hspace{0.5cm} \forall \alpha_1,\alpha_2\geq\alpha_0$$

Rizwan Ahmed
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  • @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 08:58
  • (1) A Banach space is, by definition, a complete metric space, using the metric induced by the norm. (2) Consult the reference given by @user161825. – triple_sec Aug 13 '14 at 08:59
  • @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 09:03
  • @triple_sec thanks for the help – Rizwan Ahmed Aug 13 '14 at 09:09
  • @RizwanAhmed You're welcome! – triple_sec Aug 13 '14 at 09:10

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