Q: Sequence of Pointwise bounded bounded and equicontinous functions is uniformally bounded The problem i have today is that let K be a compact space and fn   be a sequence of pointwise bounded and equicontinous real or complex valued functions then this sequence is uniformally bounded (without using arzela ascoli ) ...what i was thinking that K is compact so totally bounded but together with the given conditions i could do something but i got stucked so any help ?or hints how to develop those arguments ?

A: Hint: 
Let $\epsilon>0$. Pick a $\delta>0$ from equicontinuity. The collection $\{B(z,\delta)\}_{z \in X}$ is an open cover. If $z_1, \cdots, z_n$ correspond to a finite subcover, use pointwise boundedness on these fintely many points.

A: Hint: Each $x \in K$ has a neighbourhood $U(x)$ such that for $y \in U(x)$ and any $n$, $|f_n(y) - f_n(x)| < 1$.  Take a finite subcover...
