## Winter 1995

Problem Statement: Let $f_k:D\rightarrow\mathbb{R}, D\subseteq\mathbb{R}$ and let $|f_k(x)|\leq M_k$ for all $k\in\mathbb{N}$ and for every $x\in D$. Show that if $\sum\limits_{k=1}^{\infty}M_k$ converges then $\sum\limits_{k=1}^{\infty}f_k$ converges uniformly on $D$.

Note: This is asking us to prove the Weierstrass M-Test.

Proof: Let $f_k:D\rightarrow\mathbb{R}$ and $|f_k(x)|\leq M_k$ for every $k\in\mathbb{N}$ and for every $x\in D$. Assume that $\sum\limits_{k=1}^{\infty}M_k$ converges. This implies that the sequence of partial sums, $\{T_K=\sum\limits_{k=1}^{K}M_k\}$, converges. Every convergent sequence is Cauchy.

We wish to show that for every $\varepsilon>0$ there exists $K\in\mathbb{N}$ such that for all $n,m\geq K$ and for every $x\in D$ it follows that $|S_n(x)-S_m(x)|<\varepsilon$. Note that $S_N(x)=\sum\limits_{k=1}^{N}f_k(x)$.

Let $\varepsilon>0$. Then there exists a $K\in\mathbb{N}$ such that for $n,m\geq K$ it follows that $|T_n-T_k|<\varepsilon$. This implies that for $n,m\geq K, |\sum\limits_{k=m}^{n}M_k|<\varepsilon$.

Let $n,m\geq K$ and without loss of generality suppose that $n>m$. Then it follows that:

$|S_n(x)-S_m(x)|=|\sum\limits_{k=m}^{n}f_k(x)|$

$\leq |M_m+M_{m+1}+\dots+M_n|$ for every $x\in D$

$=|\sum\limits_{k=m}^{n}M_k|<\varepsilon$

Thus, $\sum\limits_{k=1}^{\infty}f_k(x)$ converges uniformly.

$\Box$

Reflection: I’m really glad I did this proof. This has helped me remember the Weierstrass M-Test so I can use it on the qual if I need to, but it’s also a really good technique to know. When trying to show something converges uniformly you just need to show that the sequence of partial sums is Cauchy for every $x$ in your domain.