Let \(\{x_n=(x_{n1},x_{n2}, \cdots,) \}\)be a sequence in the product space \(\Pi_{\alpha \in I} X_\alpha\) with product topology. Then the sequence \(\{x_n\}\) converges to\(x+(x_1,x_2, \cdots,)\in \Pi_\alpha X_\alpha\) iff for every positive integer \(m, \{\pi_m(x_n)=x_{nm} \}\) converges to \(x_m\).

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