I want to know if the following proof is correct. Let $M,N$ and $X$ be random variables. If the following inequality holds for any concave non-decreasing function $u$ \begin{equation} \mathbb{E}[u(N)]\leq\mathbb{E}[u(M)] \end{equation} Then \begin{equation} \mathbb{E}[u(N+X)]\leq\mathbb{E}[u(M+X)] \end{equation} My attempt: Suppose that $u´\leq C$ for some constant $C$. Given that $u$ is non-decreasing $u'\geq0$ and $u$ is concave, then \begin{align*} \mathbb{E}[u(N+X)-u(M+X)]&\leq \mathbb{E}[u'(M+X)(N-M)]\\ &\leq C\mathbb{E}[N-M]\\ &= C(\mathbb{E}[N]-\mathbb{E}[M])\\ &\leq 0 \end{align*} where the last inequality holds by the hypothesis using $u(x)=x$. Given that $C$ is arbitrary then is possible to conclude that for any $u$ (not necessary bounded) $\mathbb{E}[u(N+X)]\leq\mathbb{E}[u(M+X)]$.
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