Theorem: Let be a sequence of bounded on functions and with bouned. Then, is uniformly bounded.
Proof: We know that if and only if with the norm. So, choose such that . Then, let . Then, let be arbitrary. If then evidently if then . Regardless, for every we have that . Or, that for every and every we have that .
Remark: The above proof is not necessarily the most general but will work for our purposes.
Thus, noticing that the continuity of and compactness of guarantees that each is bounded, and since the uniform limit of continuous functions is continuous we see that is a continuous function on a compact space and thus also bounded. Thus, the theorem applies