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## Homework Statement

I really got a problem with these products.

If Xn is metrizable with dn, and if D(x, y) = suo{di(xi, yi)/i} is the metric which induces the product topology on X = ∏ Xn, show that if Xn is totally bounded for every n (under dn), then X is totally bounded under D. Conclude without using the Tychonoff theorem that a countable product of compact metrizable spaces is compact.

## The Attempt at a Solution

Since Xn is totally bounded for every n, this means that for any n and any ε > 0 there exists a finite covering of Xn by ε-balls. Somehow, I must find a finite collection of ε balls (if ε is given) in X which covers X. Any discrete hint is welcome... I had a few ideas, but they don't work.