simple question on bijection

This seems trivial, but I'm getting stuck/confused how to prove it formally. Please help.

Consider expansions of

$\displaystyle (1+x+...+x^{a_1})(1+x+...+x^{a_2})....(1+x+...+x^{ a_n}) $

Let the coefficient of $\displaystyle x^N$ be $\displaystyle C$

Let me define a set, $\displaystyle S$ as $\displaystyle S=\{(y_1,y_2,...y_n)|x^{y_1+y_2..+y_n}=x^N\}$

Here $\displaystyle y_i \in {0,...,a_i}$

I want to show that

$\displaystyle S\cong\{1,2,..,C\}$

I know what's going on behind the scenes, but just not able to express it coherently/rigorusly. I mean I'm not getting a smart bi-jection between the two sets