Solutions

Let $I_1,I_2,\ldots$ be the sequence of intervals formed by iteratively picking the unbounded half. Note that we can always find an unbounded half because otherwise the current interval would be bounded.

Let

$$y\in\bigcap_{i=1}^{\infty} I_i.$$

Since $f$ is continuous on $[a,b]$ and $y\in[a,b]$, then $f$ is also continuous at $y$. By Theorem 1, there exists $\delta>0$ such that $f$ is bounded for $x\in(y-\delta,y+\delta)$. As in the previous exercise, we know that the length of our intervals is

$$\frac{b-a}{2^n}.$$

Hence, there exists $n\in\mathbb{N}$ such that $y\in I_n$ and

$$\frac{b-a}{2^n}<\delta.$$

But this means that $I_n$ is bounded, which contradicts the definition of those intervals.