Combination Theorem for Sequences/Complex/Sum Rule/Proof 1

Theorem
Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.

Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:


 * $\ds \lim_{n \mathop \to \infty} z_n = c$
 * $\ds \lim_{n \mathop \to \infty} w_n = d$

Then:

Proof
Let $\epsilon > 0$ be given.

Then $\dfrac \epsilon 2 > 0$.

We are given that:
 * $\ds \lim_{n \mathop \to \infty} z_n = c$
 * $\ds \lim_{n \mathop \to \infty} w_n = d$

By definition of the limit of a complex sequence, we can find $N_1$ such that:
 * $\forall n > N_1: \cmod {z_n - c} < \dfrac \epsilon 2$

where $\cmod {z_n - c}$ denotes the complex modulus of $z_n - c$.

Similarly we can find $N_2$ such that:
 * $\forall n > N_2: \cmod {w_n - d} < \dfrac \epsilon 2$

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:
 * $\ds \lim_{n \mathop \to \infty} \paren {z_n + w_n} = c + d$