Let $X_1$, $X_2$, $\dots$ be an i.i.d. sequence of independent random variables with finite mean and $N$ a non-negative integer-valued random variable with finite mean, independent of $X_i$ for all i. Let $S_k = X_1 + \dots + X_k$.
It's pretty easy to prove the Wald's equation $\mathbb{E}[S_N] = \mathbb{E}[N]\mathbb{E}[X_1]$, but before I want to prove that $S_N$ has finite mean:
$$\sum_{x}|x|\mathbb{P}(S_N=x) = \sum_{x} \sum_{k=0}^{\infty}|x|\mathbb{P}(S_k=x, N= k) = \sum_{x} \sum_{k=0}^{\infty}|x|\mathbb{P}(S_k=x)\mathbb{P}( N= k) = \sum_{k=0}^{\infty}\mathbb{P}( N= k)\sum_{x}|x|\mathbb{P}(S_k=x)$$
for all fixed $k$ the series $\sum_{x}|x|\mathbb{P}(S_k=x)$ converges because $S_k$ has finite mean but why I can say that the whole series converges?
$\sum_{x}|x|\mathbb{P}(S_k=x)=E|S_k|\leq E|X_1|+E|X_2|+\cdots+E|X_k|=kE|X_1|$. Finally, $\sum P(N=k)\sum_{x}|x|\mathbb{P}(S_k=x)\leq \sum P(N=k) kE|X_1|=(E|X_1|)(EN)<\infty$