As the Catalan numbers are defined as $$C_n = \frac{1}{n+1} \binom{2n}{n},$$ it is not immediately clear that they are integers.
To show that they are, there's a relatively basic approach involving some binomial identities, but I wanted to avoid most of these, so I tried the following.
To show that $C_n$ is integer, it obviously suffices to show that $n + 1 \mid \binom{2n}{n}$. Given $n$, we can see that $$ \binom{2n}{n+1} = \frac{n}{n+1} \binom{2n}{n} $$ by manipulating the fractions a little. Thus, $$ \binom{2n}{n+1} (n+1) = n \binom{2n}{n},$$ and therefore $n + 1 \mid n \binom{2n}{n}$. Since $n$ and $n+1$ are coprime, $n+1 \mid \binom{2n}{n}$, which should complete the proof.
Is this proof correct? For some reason, it feels like there's something off with it, although I can't see any mistakes.