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
$$|F(x) - F_n(x)| = |F(x) - S + S - S_n + S_n - F_n(x)|$$
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 $\epsilon > 0$ 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
$$|F(x) - F_n(x)| = |F(x) - S + S - S_n + S_n - F_n(x)|$$
and recall the definition of convergence of $S_n \to S$.

Scratch that: instead consider that, for all x,
$$|F(x) - F_n(x)| = \sum_{k=n+1}^{\infty} f_k(x) \leq \sum_{k=n+1}^{\infty} s_k = |S - S_n|$$
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.