I would like to show that a countable product of sequentially compact spaces is a sequentially compact space.
To show that a product of N sequentially compact spaces is sequentially compact, I think that we can take a sequence $(x_1, x_2, \cdots, x_N)$ in the product of sequentially compact spaces, and say that each sequence $x_1, x_2, \cdots, x_N$ is in the sequentially compact space $X_i$, so has a convergent subsequence. Then the global sequence converges to the sequence of limits of the sequences $x_1, x_2, \cdots, x_N$.
I don’t understand why this reasoning would not work in the case of an infinite product space and how to prove, then, the result.