I'll summarize this question as follows:

Suppose (fn) is a an equicontinuous sequence of functions in C0[M, R]. Suppose p is a point in M such that fn(p) is a bounded sequence of real numbers. What is the condition on M so that (fn) is uniformly bounded (that is, sup {lfn(x)l} <= C for some C}



The condition I got is that M must be totally bounded.

Here's why:

Given: lfn(p)l <= M for all n

Since (fn) is equicontinuous, then there exists a L such that:

d(x,y) < L --> lfn(x) - fn(y)l < 1

Since M is totally bounded, there are finitely many points in M, call them {x1, x2, ... xk} where M is in the union of the open balls of radius L/2 around these points.

Fix x in M. Look at lfn(x) - fn(p)l

We know x is in the open ball around xi and p is in the open ball around xj (suppose, WLOG, j>i)

Thus, lfn(x) - fn(p)l <= lfn(x) - fn(xi)l + ... + lfn(xj) - fn(p)l

but d(x, xi) < L, d(xi, x(i+1))<L and d(xj, p) < L

--> lfn(x) - fn(p)l <=(j-i+1)*1 <= k

so lfn(x)l <= k + lfn(p)l <= k+M = C

...thoughts? (Relating to the proof, not the lack of latex)