Proving that a sequence is bounded without knowing the sequence explicitly

Question

Given that $f(x)=x^2+\frac{1}{4}$, there exists the iterated sequence ${f^{\circ n}(x)}_{n=1}^\infty$ (where $f^{\circ n}(x)$ is defined as $\underbrace{f(f(f...(x)...))}_{n\text{ times}}$), which is increasing on the interval $(-1/2, 1/2)$. I need to prove that this sequence is bounded on this interval, without actually knowing the sequence explicitly.

Can you please give me a hint as to how such a proof would work? I still have no idea, besides the fact that it is possible to do the proof by just using the function $f(x)$ above.

Answer 1

Notice that $f(x) \leq \frac{1}{2}$ for all $x \in (-\frac{1}{2},\frac{1}{2})$ [you can do this by noting that f is increasing on $[0,1/2)$]

Now suppose that $f^{\circ n}(x)<\frac{1}{2}$ for all $x \in (-1/2,1/2)$. Then show how this implies that $f^{\circ n+1}(x)<\frac{1}{2}$.

By induction...

Answer 2

Hint: By symmetry just consider $x\in[0,\frac{1}{2})$. Now notice that $f([0,\frac{1}{2}))=[0\frac{1}{2}).$