A sequence $(f_n)$ of bounded functions on $A \subseteq \Bbb{R}$ converges uniformly to $f$ if and only if $||f_n - f||_A \to 0$.
I was working the through the proof of the above theorem when I noticed, as it appeared to me, that the boundedness assumption was never explicitly invoked. My question is, exactly where is this assumption used? I give a screenshot of the proof below:
EDIT:
The norm $||\cdot||_A$ is defined as $||f||_A := \sup \{|f(x)| ~|~ x \in A\}$, where $f$ is some function from $A$ to $\Bbb{R}$.
This is quite simple. Usually you would choose a sequence of continuous functions and $A \subseteq \mathbb R$ might be compact. Then you can define the seminorm $\Vert f \Vert_A = \max_{x \in A} \vert f(x) \vert$ without any doubt since continuous functions have a maximum on a compact sets. In your case none such properties are given. So if you want to define the seminorm $\Vert f \Vert_A = \sup_{x \in A} \vert f(x) \vert$ you need to make sure that it is well defined, i.e. $\Vert f \Vert_A \neq \infty$. The most simple way to do that is to demand that your sequence is bounded on $A$. In this case the seminorm is always well-defined and you can use it in your proof :)