How to show that a disjoint sum of metric spaces is metrizable?

How to show that a disjoint sum of metric spaces is metrizable?

I have encountered this question while tring to figure out why every
sequential space $X$, is a quotient spaces of a metric space. If I
understand correctly, given a sequential space, every sequence (including
it's limit) can be identified with the space $Y=\{0\} \cup \{\frac1{n+1}|n
\in N\}$. So The space $X$ can viewed as a quotient space of
$\bigoplus_{(x_n) \in C} \{(x_n) \} \times Y$ where $C$ is the set of all
converges sequences in $X$. My question is, How can I show that this space
is metrizable?
Thank you! Shir