Is it true that $0<0$ is a part of the proof for a unique fixed point of a contraction map?

Question

I think it isn't, but my friend told me that this statement is true, he said that $0<0$ is even part of the proof for a unique fixed point of a contraction map.

I google it but I couldn't find anything.

Here is the proof according to him:

Take a contraction map $T$, so $d(Tx,Ty)≤qd(x,y)$ where $q\in[0,1)$. Now assume there are two fixed points, say $x_1,x_2$. Then we have: $$0\leq d(x_1,x_2)=d(Tx_1,Tx_2)\leq qd(x_1,x_2) ⇒0\leq(1−q)d(x_1,x_2)\leq0$$ But this can hold only if $d(x_1,x_2)=0$, which implies $x_1=x_2$. But that also gives us: $$d(x_1,x_2)<d(x_1,x_2)⇒ 0<0 $$

Answer 1

It is not true ("that $0<0$ is a part of the proof for a unique fixed point of a contraction map"). Your proof stops at "$x_1=x_2$." Anything after that is probably gibberish that drops the equality part of $\leq$ for no good reason. (I wouldn't know, I'm ignoring that wrong part at the end.)