So I was reading this lemma which states: Let $m,n$ be natural numbers such that $1 \leq m \leq n$. Then \begin{equation*} {n\choose m-1} + {n\choose m} = {n+1\choose m}. \end{equation*} It follows from this lemma using induction that the binomial coefficients are integers, rather than just rational numbers.
Using induction I could only prove that this equation is true for all natural numbers. How would I show that these coefficients are integers?
I suppose that, in this context, $\binom nm$ is defined as$$\binom nm=\frac{n!}{m!(n-m)!}.$$With this definition, it is clear that $\binom nm\in\mathbb Q$, but it is not clear that it is an integer.
However, it is clear that $\binom n0=\binom nn=1$, which is an integer. And, since$$\binom n{m-1}+\binom nm=\binom{n+1}m,$$it follows (by induction on $n$), that each $\binom nm$ is an integer.