How do you show that the fixed point is unique?
I would rather have hints and tips so that I can try and work through it! But I just don't know where to start!
A contraction is a function $\displaystyle f$ such that $\displaystyle d(f(x),f(y))\leq kd(x,y)$ for $\displaystyle k<1$.
Say that $\displaystyle x_0$ is a fixed point (so that $\displaystyle f(x_0)=x_0$). Assume that it is not unique; that is, there is another fixed point $\displaystyle y_0$.
Now look at $\displaystyle d(x_0,y_0)$. What can you say?
I used proof by contradiction.
Let phi: A->A be a contraction with constant k<1
Assume that there exists 2 fixed points
phi(x1) = x1
phi(x2) = x2
According to the definition
|phi(x1) - phi(x2)|<=k|x1-x2|
as phi(x1) = x1 and phi (x2) = x2 then
|x1-x2|<=k|x1-x2|
which gives
1<= k
which contradicts the original condition that k<1....
Am i fumbling along the right lines, or is it all rubbish?