Convergence and Uniform Convergence of Sequences of Functions

[Yagoda]

Homework Statement

f_n is a sequence of functions and s_n is a sequence of reals such that 0 ≤ f_n(x) ≤ s_n for all x.
I want to show that if \sum_{k=0}^{n}s_k is Cauchy then \sum_{k=0}^{n}f_k is uniformly Cauchy and that if \sum_{k=0}^{\infty}s_k converges then \sum_{k=0}^{\infty}f_k converges uniformly.

Homework Equations

The Attempt at a Solution

If \sum_{k=0}^{n}s_k is Cauchy then that means there exists an N such that \left|\sum_{k=0}^{n}s_k-\sum_{k=0}^{m}s_k\right|<\epsilon for all \epsilon where m,n >N.
Also \sum_{k=0}^{n-1}f_k\leq \sum_{k=0}^{n}f_k for all n because every f_n is at least zero and \sum_{k=0}^{n}f_k\leq \sum_{k=0}^{n}s_k.
I guess I'm missing how to put these pieces together.

 

[pasmith]

There's a fact about real Cauchy sequences which you should know: a real sequence converges if and only if it is Cauchy (if you don't know that, try to prove it for yourself).

If \sum s_n converges, then \sum f_n(x) converges for all x, because for all x every term is positive and less than or equal to the corresponding term of \sum s_n.

Just for convenience I'll define S = \sum_{k=0}^{\infty} s_k, S_n = \sum_{k=0}^n s_k, F(x) = \sum_{k=0}^{\infty} f_k(x) and F_n(x) = \sum_{k=0}^n f_k(x).

You want to show that F_n \to F uniformly, the definition of which is that for all \epsilon > 0 there exists N \in \mathbb{N} such that for all x, if n \geq N then |F(x) - F_n(x)| < \epsilon. So you might like to consider
<br /> |F(x) - F_n(x)| = |F(x) - S + S - S_n + S_n - F_n(x)|<br />
and recall the definition of convergence of S_n \to S.

You may then want to satisfy yourself that if F_n \to F uniformly then F_n is uniformly Cauchy (and vice versa).

 

[pasmith]

You want to show that F_n \to F uniformly, the definition of which is that for all \epsilon &gt; 0 there exists N \in \mathbb{N} such that for all x, if n \geq N then |F(x) - F_n(x)| &lt; \epsilon. So you might like to consider
<br /> |F(x) - F_n(x)| = |F(x) - S + S - S_n + S_n - F_n(x)|<br />
and recall the definition of convergence of S_n \to S.

Scratch that: instead consider that, for all x,
<br /> |F(x) - F_n(x)| = \sum_{k=n+1}^{\infty} f_k(x) \leq \sum_{k=n+1}^{\infty} s_k<br /> = |S - S_n|<br />
and recall the definition of convergence of S_n \to S.

 

[Yagoda]

That was very helpful. Thanks for reminding me about the relationship between Cauchy and convergence of real sequences. Don't know how I overlooked that.