Prove that if both $a$ and $b$ divided by $n$ give remainder 1, then $ab$ divided by $n$ gives remainder 1.

Yes this is good. One could also use modular arithmetic to say that $a\equiv 1\pmod n$ and $b\equiv 1\pmod n$ gives $ab\equiv 1\cdot 1\equiv 1\pmod n$, however I like your approach better because it uses the actual remainder as in the division algorithm, rather than using the relationship between remainders and modular arithmetic.