Consider a sequence $(\bar{x_n})_{n \in \mathbb{N}}$ defined on an infinite dimensional space, with e.g the product topology defined on that space. Is it then true that (omitting the ${n \in \mathbb{N}}$ subscript):
$(\bar{x_n}) \underset{n \rightarrow \infty}{\rightarrow} x$ in the product topology $\iff$ for each component we have $(\bar{x_n}_j) \underset{n \rightarrow \infty}{\rightarrow} (\bar{x_j})$, where $j$ corresponds to the dimension/component?
Is this equivalence true? If so, I am assuming it is a special case for when the product topology is defined on the infinite dimensional space?
My confusion really is when can one check what a sequence (defined on an infinite dimensional space) converges to by simply looking at the convergence component wise? Because for finite dimensional spaces I am aware that the pointwise convergence limit (i.e component wise convergence limit) will be the same as the uniform convergence limit on the product space. But for infinite dimensional spaces, as http://www-groups.mcs.st-andrews.ac.uk/~john/analysis/Lectures/L17.html points out with a good example, this is generally not possible. Hence, back to the above statement, is it true? Please let me know if something is not clear.