summation - Limit of triple sum

Suppose one has the following triple sum:

$$S_n=\sum_{s=0}^n\sum_{t=0}^s\sum_{u=0}^sf(n,t)g(n,u)$$

where for all $n$, $-\alpha< S_n <\alpha$ for some real constant $\alpha<\infty$. Since $S_n$ is bounded above and below by a constant may one interchange the limit with the first summand, obtaining
$$\lim_{n\to\infty}S_n=\sum_{s=0}^{\infty}\lim_{n\to\infty}\left(\sum_{t=0}^s\sum_{u=0}^sf(n,t)g(n,u)\right)?$$

Since the limit is now inside the first summand, may one now consider $s$ as a constant and thus bring the limit inside the two other summands to the right of it, yielding

$$\lim_{n\to\infty}S_n=\sum_{s=0}^{\infty}\sum_{t=0}^s\sum_{u=0}^s\left(\lim_{n\to\infty}f(n,t)g(n,u)\right)?$$

If not, why not?