Prove the Wald's identity using that $E[S_{N}]=M^{\prime}_{S_{N}}(0)$

Question

Let $X_{1}, X_{2}, \ldots$ i.i.d., and $N$ be an independent variable with positive integer values. Let $S_{N}=$ $X_{1}+\cdots+X_{N}$ be the random sum. Let us assume that the moment-generating functions of $X_{1}$ and $N$are finite in some interval around the origin.

My attempt:I want to prove that: $$ E[S_{N}]=E[N]E[X_{1}] $$ I have: $$ M_{S_{N}}(t)=M_{N}(\log M_{X_{1}}(t)) $$ Then: $$ M_{N}(\log M_{X_{1}}(t))=E[e^{\log M_{X_{1}}(t)N}] $$

I need to use series $e^x =\sum_{k=0}^\infty \frac{x^k}{k!}$ and replace or there is another way?

Answer 1

$$ M_{S_N}'(0)=\frac{d}{d t}\left(\mathsf{E}[M_{X_1}(t)]^N\right)|_{t=0}=\mathsf{E}\!\left[N M_{X_1}(0)^{N-1}\times M_{X_1}'(0)\right]=\mathsf{E}N\times \mathsf{E}X_1. $$