Corollary 2

Let $f = (f_i)$ be a mapping of a set $X$ into a product space $Y = \prod_{i \in I} Y_i$. Then $f$ has a limit $y = (y_i)$ with respect to a filter $\mathfrak{F}$ on $X$ if and only if for each $i \in I$ $f_i$ has limit $y_i$ with respect to $\mathfrak{F}$.