Sequences, series, summability Proposition

Limit of a quotient of sequences

Builds on Limit of a sequence Limit of a product of sequences

Statement

Let $(a_n)$ and $(b_n)$ be real sequences, and let $a$ and $b$ be real numbers with $b \ne 0$ . Assume that $a_n \to a$ and $b_n \to b$ as $n \to \infty$ . Then there is a natural number $N$ such that $b_n \ne 0$ for all $n \ge N$ . Moreover, the sequence $(a_n / b_n)_{n \ge N}$ converges to $a / b$ .

Remarks

The main step is $1/b_n \to 1/b$ . For large $n$ we have $|b_n| > |b|/2$ , and then $|1/b_n - 1/b| = |b - b_n| / (|b_n|\,|b|) < 2 |b_n - b| / |b|^2$ . The limit of a product of $(a_n)$ and $(1/b_n)$ then gives the result. The hypothesis $b \ne 0$ is needed. For $a_n = 1$ and $b_n = 1/n$ , the quotient is $n$ , which has no real limit.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Sequence definition
Real sequence definition
Bounded sequence definition
Limit of a sequence definition
Convergent sequence definition
Epsilon-N definition of the limit of a sequence statement
Convergent sequences are bounded statement
Limit of a product of sequences statement
Limit of a quotient of sequences statement

Where this leads

13 pages build on this one.

Tag	`03S6GWWKEZ`
URI	`m2://03S6GWWKEZ`
Kind	statement, proposition
Writing	W0 stub
MSC	40A05
Status	This copy of the site has no build data. Run `m2 status 03S6GWWKEZ`.
Uses	Limit of a sequence, Limit of a product of sequences

Limit of a quotient of sequences

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

Used by