---
tag: 0RAM4E07NR
kind: statement
role: proposition
name: Limit of a sum of sequences
msc: [40A05]
uses: [0XKWSSY8P2, 09G6KHS6GY]
sources:
  - url: https://en.wikipedia.org/wiki/Limit_of_a_sequence
    locator: Properties
---

## 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|0734RSVF2V]], it also gives $a_n - b_n \to a - b$.
It is one of the rules of the [[algebra of limits|07XBM3A9J5]].

## Explanations

### L1 human

If $a_n$ is close to $a$ and $b_n$ is close to $b$, then $a_n + b_n$ is close to $a + b$.
The two errors add, so we make each error smaller than half of the allowed error.