Dirichlet's Approximation Theorem for integer

Question

This theorem says that if $\alpha$ is a real number and $n$ is a positive integer, then there exist integers $a$ and $b$ with $1\leqslant a\leqslant n$ such that $\vert a\alpha - b\vert < 1/n$.

My question is: how can the theorem be true when $\alpha$ is an integer (and it could be an integer, if it's a real number...)? In this case it seems it can't be less then $1$.

Answer 1

If $\alpha$ is an integer, you can just take

$$\begin{cases} a=1 \\ b=\alpha.\end{cases}$$

Then you have

$$\vert a\alpha -b\vert =0 <\frac 1n$$

for all $n\geqslant 1$.