- #1

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Let $$\{X_{\alpha}\}_{\alpha \in I}$$ a collection of topological spaces and $$X=\prod_{\alpha \in I}X_{\alpha}$$ the product space. Let $$p_{\alpha}:X\rightarrow X_{\alpha}$$, $$\alpha\in I$$, be the canonical projections

a)Prove that a sequence $$\{a_n\}$$ converges on $$X$$ if and only if the sequence $$\{p_{\alpha}(a_n)\}$$ converges on $$X_{\alpha}$$ for all $$\alpha \in I$$.

b) Let $$I$$ the set of all sequences $$\alpha:\mathbb{Z}_{\geq 1}\rightarrow \{-1,1\}$$. Let the sequense $$a_n=\prod_{\alpha \in I}\alpha(n)\in [-1,1]^{I}$$. Prove that $$\{a_n\}$$ does not have a convergent subsequence. Is $$[-1,1]^{I}$$ sequentially compact? Is $$[-1,1]^{I}$$ firts contable?

My attempt:

a) Take $$I = \mathbb{N}$$ and $$X_i = \mathbb{R}$$ for all $$i \in \mathbb{N}$$. Now the elements in $$X$$ are real sequences and the goal is to prove that if $$a_n$$ is a sequence of these sequences (i.e. a bi-infinite real sequence), then it converges if and only if the real sequence $$a_n^{(k)}$$ converges for all $$k \in \mathbb{N}$$.

How would the demonstration of the general case be?

b) I don't know