Is there a name for this theorem about the convergence of a function?

Question

Let $f(x)$ be a continuous function over $\mathbb{R}$ such that for all $a < b$, we have $a < f(a) < b$. Then, for any $x < b$, the sequence $\{t_n\}$ defined by $t_0 = x, t_n = f(t_{n-1})$ converges at $b$.

I want to cite the above proposition in an assignment I'm writing, but I don't know what name to give it, and I don't want to rewrite it down individually every time I use an assignment.

Answer 1

Firstly, it took me a few times to parse your theorem correctly. I would probably state it as

Theorem. Let $(a,b]\subset \mathbb{R}$ be a fixed interval where $a$ can be $-\infty$. If $f:(a,b]\to \mathbb{R}$ is a continuous function such that for every $x\in (a,b)$, $x < f(x) < b$, then the sequence etc.

(if $b$ were allowed to vary, your statement is vacuous. So it is important to make it clear in your quantifiers that $b$ is fixed.)

Secondly, continuity is unnecessary in your theorem. From the statement that $x < f(x) < b$ for all $x < b$ where $b$ is fixed, you have that the sequence $(t_n)$ is a bounded monotonic sequence of real numbers and hence must converge. This fact is usually cited as "by the convergence of bounded monotonic real sequences etc."

But the real question is where are you going with this?

1. Since you mentioned continuity, your result may be interpreted as a variant of Brouwer's fixed point theorem.
2. One can alternatively think of your result as some corollary of Cantor's intersection theorem.
3. If you have some quantified rates on how much larger $f(x)$ is compared to $x$, then you may be able to call it a version of the contraction mapping principle.

Lastly, I don't think this theorem really needs a name. If you must refer to it: write it down once, call it Theorem 1, and just say, "By Theorem 1..."