Limit of a sum of sequences

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.

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.