I have come across a proof which I understand almost completely, except for one part:
THEOREM: If $f$ is uniformly continuous on a bounded interval $I$, then $f$ is also bounded on $I$.
PROOF: In this case we assume that $I$ is of the form $(a,b), (a,b], [a,b)$, or $[a,b]$, with $a,b \in \mathbb{R}$. Fix an $\epsilon > 0$, for instance $\epsilon = 1$. Since $f$ is uniformly continuous, there is a $\delta > 0$ such that:
$|f(x_1) - f(x_2)| < \epsilon = 1$ when $x_1, x_2 \in I$ and $|x_1 - x_2| < \delta$
Divide $I$ into $N$ intervals, $I_1, . . ., I_N$, where $N$ is chosen so that $\frac{b-a}{N} < \delta$.
Let $z_i$ be the center point of $I_i$. For each $i$ and $x \in I_i$, $|x - z_i| < \delta$, and then we have:
$|f(x)| = |f(x) - f(z_i) + f(z_i)| \leq |f(x) - f(z_i)| + |f(z_i)| \leq 1 + |f(z_i)|$. Then for $x \in I_i$,
$|f(x)| \leq 1 + \max_{1 \leq i \leq N}\{|f(z_i)|\}$.
Let $M = \max_{1 \leq i \leq N}\{|f(z_i)|\}$. Then $|f(x)| \leq 1 + M$
QED
OK, so the one thing I am a bit unsure of here, is when we write:
Let $M = \max_{1 \leq i \leq N}\{|f(z_i)|\}$.
How is it that we know for sure that each $|f(z_i)|$ is also bounded? I see how the presence of a maximum value completes the proof, but why is it not possible that we have an $|f(z_i)|$ which is unbounded?
If anyone could explain this to me I would greatly appreciate it!
