I conjecture, and would like to prove: A convergent sequence specified recursively by a continuous function $f$ converges to a fixed point of $f$. That is, $f$ must have at least one fixed point, one of which is the limit of the sequence.
First, let's state this conjecture formally:
Let $(x_n)$ be a sequence such that $x_{n+1} = f(x_n)$. I conjecture that if $(x_n)$ converges to a limit $\ell$, and $f$ is continuous, then $f(\ell) = \ell$.
(Continuity is clearly required, as evidenced by $x_n = 1, f(x) = \{x/2$ for $x > 0,$ otherwise $1\}$.)
Proof: Consider the sequence $(F_n)$, such that $F_1 = x_1$ and $F_{n+1} = f(x_n)$. Since, $F_n = x_n$,
$$ \ell = \lim_{n \to \infty} F_n = \lim_{n \to \infty} f(x_n) $$
Since $f$ is continuous, $$\lim_{n \to \infty} f(x_n) = f(\lim_{n \to \infty} x_n) = f(\ell).$$
Thus, $$ \ell = f(\ell),$$
QED.
Is this proof correct? Can any steps be omitted? It seems to me more complex than I expected, yet I can't find a step that is superfluous.