Combination Theorem for Sequences/Real/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 $\dfrac \epsilon 2 > 0$.

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

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

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

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

Thus $\forall n > N$:

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