Combination Theorem for Sequences/Sum Rule

Theorem

Let $X$ be one of the standard number fields $\Q, \R, \C$.

Let $\left \langle {x_n} \right \rangle$ and $\left \langle {y_n} \right \rangle$ be sequences in $X$.

Let $\left \langle {x_n} \right \rangle$ and $\left \langle {y_n} \right \rangle$ be convergent to the following limits:

$\displaystyle\lim_{n \to \infty} x_n = l, \lim_{n \to \infty} y_n = m$

Then:

$\displaystyle\lim_{n \to \infty} \left({x_n + y_n}\right) = l + m$

Proof

Let $\epsilon > 0$ be given. Then $\frac \epsilon 2 > 0$.

Since $\displaystyle\lim_{n \to \infty} x_n = l$, we can find $N_1$ such that $\forall n > N_1: \left\vert{x_n - l}\right\vert < \frac \epsilon 2$.

Similarly, since $\displaystyle\lim_{n \to \infty} y_n = m$, we can find $N_2$ such that $\forall n > N_2: \left\vert{y_n - m}\right\vert < \frac \epsilon 2$.

Now let $N = \max \left\{{N_1, N_2}\right\}$.

Then if $n > N$, both the above inequalities will be true.

Thus $\forall n > N$:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left\vert{\left({x_n + y_n}\right) - \left({l + m}\right)}\right\vert\)	\(=\)	\(\displaystyle \left\vert{\left({x_n - l}\right) + \left({y_n - m}\right)}\right\vert\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\le\)	\(\displaystyle \left\vert{x_n - l}\right\vert + \left\vert{y_n - m}\right\vert\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Triangle Inequality
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(<\)	\(\displaystyle \frac \epsilon 2 + \frac \epsilon 2 = \epsilon\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Hence $\displaystyle\lim_{n \to \infty} \left({x_n + y_n}\right) = l + m$.

$\blacksquare$

Sources

• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Proposition $1.2.11 \text {(a)}$