Proof that $0.101001000100001dots $ is irrational

If it’s rational, the for some $kgt0$, the decimal is periodic after position $k $. Say it’s periodic of period $m$.

Then there’s a $1$ appearing in every block of size $m $.

But there’s a block consisting in $n$ zeros, for $ngt m $. This implies that the decimal terminates, a contradiction.

I found out most of you took a course on writing proofs. This is textbook proof by contradiction.