Sequences, series, summability Proposition

Limit of a sum of sequences

Builds on Limit of a sequence Epsilon-N definition of the limit of a sequence

Statement

Let $(a_n)$ and $(b_n)$ be real sequences, and let $a$ and $b$ be real numbers. If $a_n \to a$ and $b_n \to b$ as $n \to \infty$ , then $a_n + b_n \to a + b$ as $n \to \infty$ .

Remarks

Given $\varepsilon > 0$ , choose $N$ so that $|a_n - a| < \varepsilon/2$ and $|b_n - b| < \varepsilon/2$ for all $n \ge N$ . The triangle inequality then gives $|(a_n + b_n) - (a + b)| < \varepsilon$ . With the constant sequence $-1$ and the product rule for limits, it also gives $a_n - b_n \to a - b$ . It is one of the rules of the algebra of limits.

Reading path

To understand this page from the ground up, read these 5 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
Limit of a sequence definition
Epsilon-N definition of the limit of a sequence statement
Limit of a sum of sequences statement

Where this leads

5 pages build on this one.

Tag	`0RAM4E07NR`
URI	`m2://0RAM4E07NR`
Kind	statement, proposition
Writing	W0 stub
MSC	40A05
Status	This copy of the site has no build data. Run `m2 status 0RAM4E07NR`.
Uses	Limit of a sequence, Epsilon-N definition of the limit of a sequence

Limit of a sum of sequences

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

Used by