Not quite.
Proof by contradiction works when we assume the opposite of what we want to proove is true. We then derive an erroneous result... a contradiction.
Our problem is... if x is irrational, prove that 1/x is irrational. The first thing we do is assume the opposite. Assume that x is irrational, but conversely, 1/x is rational. From this we will try to derive an impossibility.
A property of a rational number is that it can be written as a ratio of integers. ANY rational number can be written in the form
, where p and q are both integers.
So if we assume that 1/x is rational then:
Now multiply both sides by xq
What this statement says is that the integer q can be written as a product of the integer p, and the irrational number q. THIS is invalid, because multiplying an integer by an irrational number gives an irrational number. Hence q can't be an integer, and our initial assumption MUST be wrong.
Hence 1/x is not rational. And any number that is not rational is irrational.