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

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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:

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


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 $\forall n > N$:

\(\ds \cmod {\paren {z_n + w_n} - \paren {c + d} }\) \(=\) \(\ds \cmod {\paren {z_n - c} + \paren {w_n - d} }\)
\(\ds \) \(\le\) \(\ds \cmod {z_n - l} + \cmod {w_n - m}\) Triangle Inequality for Complex Numbers
\(\ds \) \(<\) \(\ds \frac \epsilon 2 + \frac \epsilon 2\)
\(\ds \) \(=\) \(\ds \epsilon\)


Hence the result:

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

$\blacksquare$