Let $X$ be a binomial random variable with parameter $\left(n,p\right)$, and let $Y$ be a Poisson random variable with parameter $np$. Let $g$ be a convex function. Prove that $$\mathbb{E}[g(X)] \le \mathbb{E}[g(Y)]$$
What is the strategy here? I thought of Jensen's inequality, and that Binomial can be approximate by Poisson. But I don't know how to proceed. The hint asked me to consider a special case where $X\sim \text{Bin}(1,p)$ and $Y\sim\text{Bin}(2, p/2)$. I worked that out, but don't see how that helps.
Hint:$^1$ generalize your special case to $$ X \sim \operatorname{Bin}(n,p), \qquad Y' \sim \operatorname{Bin}\left(k n,\frac{p}{k}\right) $$ for any arbitrary integer $k\geq 1$.
Then use the Poisson limit theorem as $k\to\infty$.
$^1$ If you try for a bit and are still stuck, leave a comment and I will fill in the details and work out a full derivation.