Calculate limit involving binomial coefficient

How can I calculate this limit. Let $p\in[0,1]$ and $k\in\mathbb{N}$.

$$\lim\limits_{n\to\infty}\binom{n}{k}p^k(1-p)^{n-k}.$$

Any idea how to do it ?

Answer

A sketch of proof: The element is
$$\frac{n(n-1)(n-2)\dots(n-k+1)}{k!} \left(\frac p{1-p}\right)^k (1-p)^n \\= A_k n(n-1)(n-2)\dots(n-k+1)(1-p)^n$$ for a certain $A_k$ which does not influence the convergence.

When $n\to\infty$ this is equivalent to
$$a_n = A_k n^k(1-p)^n$$

But $a_{n+1}/a_n = \left(\frac{n+1}{n}\right)^k(1-p) < 1-\frac p2$ as soon as $p\neq 1$, for $n$ big enough. So the limit is 0 in that case (prove it using induction).