# Homework due Jan. 23 — MTH 429H

Homework due Jan. 23:

From Rudin Ch. 7: Exercises 3, 6, 11, 12. Additional exercises:

1. (Exercise 9 from Rudin rephrased) Let $(f_n)_{n=1}^\infty$ be a sequence of continuous functions on a set $f$.
1. Prove that if $f_n$ converge uniformly to $f$ then

$\displaystyle{\lim_{n\rightarrow\infty} f_n(x_n)=f(x)}\ \ \ \ \ (\star)$

for every sequence of points $x_n\in E$ such that $x_n\rightarrow x$ and $x\in E$.

2. Is the converse true?  That is, if ($\star$) holds whenever $x_n\rightarrow x$ does it follow that $f_n$ converge to $f$ uniformly?
2. Use Theorem 7.11 from Rudin to prove the following result stated in class on Jan. 11:

Let $(a_{m,n})_{m,n=1}^\infty$ be a doubly indexed sequence such that

$\displaystyle{\lim_{n\rightarrow \infty}} a_{m,n}=b_m$

uniformly in $m$, i.e., for every $\epsilon >0$ there is an $N$ so that for $n\ge N$ we have $|a_{m,n}-b_m|<\epsilon$ for all $m$. If $\displaystyle{\lim_{m\rightarrow \infty}} a_{m,n}=A_n$ for each $n$ then $(A_n)_{n=1}^\infty$ converges and

$\displaystyle{\lim_{m\rightarrow \infty}} b_m = \displaystyle{\lim_{n\rightarrow \infty}} A_n.$