Given the norm $\| X\|_2^2 =E[|X|^2]$ and a sequence $(X_k)_k$ of random variables with $E[X_k]<0$ for each k. Assume X is a random variable such that $\|X_k -X \|_2 \overset{L^2}{\rightarrow}0$. Show that $E[X] \leq 0$.
The context for the problem is to show that $E[X]$ provides a coherent risk measure according to the aiomx given by Rockafellar in Coherent Approaches to Risk in Optimization Under Uncertainty.
My initial idea was to appeal to Cauchy-Schwarz, but I could not seem to get anywhere with that approach. My second idea was to use measure-theory by constructing the sets $A_m = \{\omega \in \Omega \ | \ X(\omega) >= X_m(\omega) \}$ for each $m$, and to check if $P(A_m) \underset{m \rightarrow \infty}{\longrightarrow} 0$. But even if I can show this, I am not convinced this shows what I wanted to show.
Is there an obvious way to approach this problem?
$E|X_k-X|^{2} \to 0$ implies that $E|X_k-X| \to 0$ and this implies that $EX_n \to EX$. Hence $EX =\lim_k EX_k \leq 0$.
Some details: $EY \leq \sqrt {EY^{2}}$ by Cauchy - Schwarz inequality. Apply this to $Y=|X_k-X|$ to justify the first step. Also $|EY| \leq E|Y|$ so $|EX_n-EX|=|E(X_k-X)| \leq E|X_k-X|$. This justifies the second implication above.