Does this method of proof work?

Question

Suppose I want to prove $a\neq b$.

I write $a=b$, simplify the expressions and eventually arrive at something like $1=2$ which is always false. Does this mean that the assumption $a=b$ is false?

Answer 1

If your proof of $1=2$ is in fact the result of the assumption $a=b$, not other assumptions, then you have used proof by contradiction and you have proved $ a\ne b.$

For example if I want to prove that $3\ne 4$ and argue that;

Let $3=4.$

Since $\sqrt 1 = -1,$ then $2=1+ \sqrt 1 = 1-1 =0$ which is false.

I have not proved $3\ne 4$ even if $2=0$ is always false.