I was thinking of doing this by contradiction. So by supposing: $$(n, n + 1) \neq 1$$
Then trying to to show that $(n, n + 1) \gt 1$ or $(n, n + 1) \lt 1$. But I'm not sure how I can accomplish this.
I also thought about the fact that $n$ and $n + 1$ are always of different parity. So if $n$ is even, then $n + 1$ is odd. But I don't think that helps me.