limit of random variables, expectation

Hallo!

Can anybody please help me?

Is the following equality true?

$\displaystyle \lim\limits_{h \to 0}\mathbb{E}[X_h]=\mathbb{E}[X]$

$\displaystyle X$ is a random variable $\displaystyle X>0$

$\displaystyle X_h$ is a sequence of random variables, $\displaystyle 1>h>0, X_h>0$

with

$\displaystyle \mathbb{E}[X_h] \leq \mathbb{E}[X]$

$\displaystyle p\lim\limits_{h \to 0}X_h=X,$ where $\displaystyle p$ denotes convergence in probability

Now, I think it follows with Fatou's Lemma for non-negative random variables, that

$\displaystyle \liminf\limits_{h \to 0}\mathbb{E}[X_h]\geq \mathbb{E}[\liminf\limits_{h \to 0}X_h]$.

Can I now say, that

$\displaystyle \lim\limits_{h \to 0}\mathbb{E}[X_h]=\mathbb{E}[X]$

is true?

Thanks in advance!