A theorem on uniform convergence

[icantadd]

Homework Statement

Theorem to be proved:
fn is uniformly convergent on E iff for every epsilon >0 there is an N, where n,m >= N then | fn(x) - fm(x) | < epsilon, for all x in E.

Homework Equations

Definition: fn is uniformly continuous on E if there is an f such that for every epsilon >0, there is an N such that for n >= N: | fn(x) - f(x) | < epsilon

The Attempt at a Solution

(only if):
Assume fn is uniformly convergent. Then
$$\forall x \in E, \forall n,m >= N, \left| f_n (x) - f(x) \right| < \frac{\epsilon}{2} and \left| f_m (x) - f(x) \right| < \frac{\epsilon}{2}$$ Thus
$$\left| f_n (x) - f_m (x) \right| \leq \left| f_n (x) - f(x) + f(x) - f_m(x) \right| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$$ Great.

Now here comes the part I wish I was doing better. I just feel like something is off. So I have two ways.

Way number 1):
Assume $$\left| f_n (x) - f(x) \right| < \epsilon \forall x \in E, \forall m,n \geq N$$ . Then fn is cauchy, and because fn is cauchy, fn is bounded on E. Then let M = max{fn(x)} x in E. Also because fn is cauchy it is convergent, and because fn <= M for all x in E, fn is convergent for all x in E. By the definition of a convergent sequence | fn(x) - f(x) | < epsilon.

Way number 2):
Assume $$\left| f_n (x) - f_m(x) \right| < \epsilon \forall x \in E, \forall m,n \geq N$$. Let f(x) be an accumulation point of \{f_n(x) : f \in E\} and, let n*>= N where $$f_{n*} \in (f(x) - \epsilon, f(x) + \epsilon) thus \left| f_{n*} (x) - f(x) \right| < \frac{\epsilon}{2}$$. Therefore

$$\left| f_n (x) - f(x) \right| = \left| f_n (x) - f_{n*}(x) + f_{n*}(x) - f(x) \right| \leq \left| f_n (x) - f_{n*}(x) \right| + \left| f_{n*}(x) (x) - f(x) \right| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \forall x \in E, \forall m,n \geq N$$

Way number two should work, but gets ugly. I don't think way number one works.

 

[VeeEight]

Assume fn is Cauchy. Then fn is bounded (easy to show). Thus it contains a convergent subsequence, call it gn. Let g = lim gn. By fn Cauchy we have |fn - fm| < e/2 and gn convergent to g implies |gn - g| < e/2
So |fn - gn + gn - g| < (or equal) |fn - gn| + |gn - g| < e