Describe the result you need.

What it says

Existence of a Bound for Continuous Functions on Compact Sets in Seminormed Additive Groups

If $s$ is a compact subset of a topological space $\alpha$ and $f : \alpha \to E$ is a function that is continuous on $s$, where $E$ is a seminormed additive group, then there exists a real constant $C$ such that $\|f(x)\| \leq C$ for all $x \in s$. This establishes that continuous functions on compact sets are bounded in seminormed additive groups.

What it says

Boundedness of Continuous Functions on Compact Sets

For any topological space $\alpha$ and compact subset $s \subseteq \alpha$, if $f : \alpha \to E$ is a function continuous on $s$ (where $E$ is a seminormed group), then there exists a real number $C$ such that $\|f(x)\| \leq C$ for all $x \in s$.

What it says

Bounded continuous function from compact space

Given a compact topological space $\alpha$ and a continuous function $f : \alpha \to \beta$ where $\beta$ is a pseudo-metric space, this constructs the corresponding bounded continuous function. The image of $\alpha$ under $f$ is compact (hence bounded) in $\beta$ by continuity, so no explicit bound is required.

What it says

Continuous Function on Compact Space is Bounded by Its Supremum Norm

For any continuous function $f$ from a compact topological space $\alpha$ to the real numbers $\mathbb{R}$, and for every point $x \in \alpha$, it holds that $f(x) \leq \|f\|$, where $\|f\|$ denotes the supremum norm of $f$, defined as the supremum of $|f(x)|$ over all $x \in \alpha$.

What it says

Boundedness of Continuous Functions with Compact Support

If a function $f : \alpha \to E$ is continuous and has compact support, then there exists a constant $C$ such that $\|f(x)\| \leq C$ for all $x \in \alpha$.