I have encounter the following difficulty in the study of limits of random variables. Assume that $\{X_n\}_{n\geq 1}$ is a sequence of real-valued random variables such that

$$\mathbb{E}[X_n]=\lambda_n,\,\, \mathbb{V}[X_n]=\sigma_n^2,$$

where, as usual, $\mathbb{E}$ and $\mathbb{V}$ are the expectation and variance. The condition that is satisfied is that $\sigma_n=o(\lambda_n)$.

Question: can we assure that the (conveniently normalized) sequence of random variables does NOT converge to a normal distribution $N(0,1)$?

As far as my intuition says a simple argument (Markov, Chebyshev,...) would be enough, but I am not able to get it.