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