Suppose $X \sim Bin(n,p)$.
How do I show that the distribution has only one peak? That is, the probability mass function of $X$ is increasing before the peak, and decreasing after the peak?
For large values of $n$, I have used the central limit theorem to show that it follows a 'bell' shape, and therefore has only one peak.
However, I am having difficulty proving the same for small values of $n$.
By "peak" I assume you mean local maximum?
The analysis will be the same as for the global maximum, though, because when you have found that
$$ \frac{ P(X = k+1)}{P(X = k)} = \left(\frac{n-k}{k+1} \right)\left(\frac{p}{1-p}\right) $$ and then find that for example that $$ \frac{ P(X = k+1)}{P(X = k)} > 1 \Leftrightarrow k < p(n+1) - 1 $$ Then this shows that $ P(X = k)$ is increasing at every $k < p(n+1) - 1$,