if $r,s$ are rational numbers, then $r+s\sqrt2$ is irrational unless $s=0$?

Question

if $r,s$ are rational numbers, Prove $r+s\sqrt2$ is irrational unless $s=0$?

I need to prove this simple question, but not sure if my method is acceptable

I'm trying to prove it by contradiction, Suppose $r+s\sqrt2 = a$, where $a$ is a rational number. Then by given $s$ not equal to $0$, we can say $\sqrt2 = \frac{a-r}{s}$ where $\frac{a-r}{s}$ is a rational number because $a,r,s$ are rational numbers, hence it must be rational, and we have a contradiction.

Did i do this correctly??

Answer 1

[Please don't upvote - just adding answer so question not stuck in unanswered state forever]

As per the numerous comments, your proof is correct.