To me it seems correct. Except for one thing: you say "divide by b" , but you're actually dividing by d. All the steps are correct though.
Quick proof veracity test for this question
Let that is not the square of another natural number. Prove that is not a rational number
Assume , hence , where and
So , so . Note , so d|a
, divide by b
But this gives a contradiction as
Hence our initial assumption is wrong and is irrational
Is this proof 100% correct?