$ \lim_{n \to \infty} \mathbb{E} \big[ -\frac{1}{n}\ln (1+e^{-nf(X_n)}) \big] = 0 $ [closed]

I want to show that the following limit is zero. $$ \lim_{n \to \infty} \mathbb{E} \big[ -\frac{1}{n}\ln (1+e^{-nf(X_n)}) \big] $$ where $(X_n)_{n \geq 1}$ is sequence of random variables, and the function $f(.)$ is positive.

Intuitively the statement should be true, but I don't know how to argue.

By Jensen's inequality $$ 0\leq E\left[\log\left(\Big(1+e^{-nf(X_n)}\Big)^{-1/n}\right)\right]\leq\log\Big(E\big[\Big(1+e^{-nf(X_n)}\Big)^{-1/n}\big]\Big) $$

Since $$\frac12\leq\frac{1}{1+e^{-nf(X_n)}}\leq 1$$ we have that $$\lim_n \Big(1+e^{-nf(X_n)}\Big)^{-1/n} =1$$

The conclusion follows from dominated convergence and the continuity of the $\log$ function.