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## Homework Statement

Let [tex](X_k,d_k)[/tex], [tex]1\leq k<\infty [/tex] be metric spaces

Let [tex]X=\prod _{k=1}^{\infty} X_k [/tex] be their Cartesian product,

that is, let [tex]X[/tex] be the set of sequences [tex](x_1,x_2,...)[/tex], where [tex]x_j\in X_j[/tex] for [tex]1\leq j < \infty [/tex]

Show that a sequence [tex] \left\{x^{(k)}\right\}_{k=1}^{ \infty } [/tex] converges in [tex]X[/tex] if and only if [tex] \left\{x_j^{(k)}\right\}_{k=1}^{ \infty } [/tex] converges in [tex] X_j [/tex] for each [tex] j \geq 1. [/tex]

## Homework Equations

## The Attempt at a Solution

Assume [tex] \left\{x^{(k)}\right\}_{k=1}^{ \infty } [/tex] converges in [tex]X[/tex].

then [tex] \left\{x^{(k)}\right\}_{k=1}^{ \infty } = (a_{11},a_{21},...),(a_{12},a_{22},...),...[/tex] where [tex] a_{ik} \in X_i [/tex]

So, [tex] \left\{x^{(k)}\right\}_{k=1}^{ \infty } [/tex] converges to [tex]c_i [/tex]

Assume [tex] \left\{x_j^{(k)}\right\}_{k=1}^{ \infty } [/tex] converges in [tex]X_j [/tex] for each [tex] j \geq 1 [/tex]

so, [tex] \left\{x^{(k)}\right\}_{k=1}^{ \infty } = (a_{11},a_{21},...),(a_{12},a_{22},...),...[/tex] where [tex] \left\{x_j^{(k)}\right\}_{k=1}^{ \infty } = a_{j1},a_{j2},...[/tex] converges to [tex]c_j[/tex]

So, [tex] \left\{x^{(k)}\right\}_{k=1}^{ \infty } [/tex] converges to [tex] (c_1,c_2,...) [/tex]