Is there a standard result out there that gives the following? It looks graphically like it must be true, but I'd like to appeal to a known result if possible.
Let $f:[0,1]\to[0,1]$ be a continuous, strictly increasing function such that $f(1)=1$ and $f(x)>x$ for all $x\in[0,1)$. For any $x_0\in[0,1)$, inductively define $x_k\equiv f(x_{k-1})$. Then $\lim_{k\to\infty}x_k=1$.
I don't know if there is a name, but here is a short proof: $x_n$ is increasing and bounded above by 1, therefore converges, $\lim x_n =\lim f(x_n)$ so $L$ must be a fixed point of the function, which means $L=1$ -- Lost1
The theorem that says the limit exists is named the monotonic convergence theorem or monotonic sequence convergence theorem... The key phrase is 'monotonic sequence'. -- Hurkyl