We know that Poisson random variable is the limiting case of a binomial random variable with parameters n and p, where n $\rightarrow \infty, p \rightarrow 0$ and $np = \lambda < \infty$ ($\lambda$ is finite).
Variance of a binomial random variable is equal to $np(1 - p)$.
So, variance of Poisson random variable
= $\lim_{p \to 0} np( 1 - p)$
= $\lim_{p \to 0} \lambda (1 - p)$
= $\lambda$
Is this kind of proof considered to be correct? If not, what is wrong with this reasoning?
This is not at all a proof, not to say a correct proof. Just because something holds for every element in a convergent sequence does not mean it also holds at the limit. For example, each element in $\frac11,\frac12,\frac13,\cdots$ is nonzero, but the limit is zero. That is what JackM means by his third point, because you cannot anyhow claim that the variance of the limit is equal to the limit of the variance. And worse still, you did not even specify what you mean by "Poisson random variable is the limiting case of a binomial random variable ...". Vague words do not constitute a proof.