Combination Theorem for Sequences/Real/Sum Rule

Theorem
Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $\R$.

Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent to the following limits:


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

Then:
 * $\displaystyle \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$

Proof
Let $\epsilon > 0$ be given.

Then $\dfrac \epsilon 2 > 0$.

We are given that:
 * $\displaystyle \lim_{n \mathop \to \infty} x_n = l$
 * $\displaystyle \lim_{n \mathop \to \infty} y_n = m$

By definition of the limit of a real sequence, we can find $N_1$ such that:
 * $\forall n > N_1: \size {x_n - l} < \dfrac \epsilon 2$

where $\size {x_n - l}$ denotes the absolute value of $x_n - l$

Similarly we can find $N_2$ such that:
 * $\forall n > N_2: \size {y_n - m} < \dfrac \epsilon 2$

Let $N = \max \set {N_1, N_2}$.

Then if $n > N$, both the above inequalities will be true:
 * $n > N_1$
 * $n > N_2$

Thus $\forall n > N$:

Hence the result:
 * $\displaystyle \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$

Also see

 * Difference Rule for Real Sequences