Recall the contraction mapping theorem:
Let $(X,\rho)$ be a complete metric space, and let $T:X\to X$ and $k\in[0,1)$ be such that$$\rho(T(x),T(y))\leq k\rho(x,y)$$for every $x,y\in X$. Then $T$ has a unique fixed point.
I was taught that if $k=1$, then a fixed point exists, but it may not be unique.
Is this true?
I think it's not since the canonical proof constructs a sequence $\{x_n\}_1^\infty$, where $x_n=T(x_{n-1})$ for $n\geq1$, and where $x_0\in X$ is arbitrary, and then shows that $\{x_n\}$ is Cauchy, which is no longer possible if $k=1$. Do we perhaps require a weaker hypothesis?