proof (1/bn) -> (1/b) from abbott Understanding Analysis.

Question

Anyone can explain this better than the book? It's difficult to understand for me.

What is the worst-case? Why does the bound on the size of $|\frac{1}{b_n}-\frac{1}{b}|$ matter?

Answer 1

Remember the definition of the limit- we want to make $|\frac{1}{b_n}-\frac{1}{b}|=\frac{|b|-|b_n|}{|b_n||b|}$ arbitrarily small, meaning for any $\epsilon>0$, there is some $N$ such that for all $n$ such that if $n\geq N$, then $\frac{|b|-|b_n|}{|b_n||b|}<\epsilon$. We know that we can make $|b|-|b_n|$ arbitrarily small, because $b_n\to b$. However, we still have to worry about $\frac{1}{|b_n||b|}$ because if it gets too big, we can't say that the whole fraction gets arbitrarily small. The proof shows that we can find an index $N_1$ to make $|b_n-b|$ small enough for our purposes, and an index $N_2$ to make $|b_n|$ big enough for our purposes. For all $n\geq\max{N_1,N_2}$, we have both sufficiently small $|b_n-b|$ and sufficiently large $|b_n|$ that we can bound $|\frac{1}{b_n}-\frac{1}{b}|$ by $\epsilon$, the details of which are all there.

That gives us that the limit of $\frac{1}{b_n}$ is $\frac{1}{b}$. Then, for any sequences $a_n\to a, b_n\to b$ with $b\neq 0$, we get that $\frac{a_n}{b_n}=a_n\frac{1}{b_n}=a\frac{1}{b}$ by (iii).