Combination Theorem for Sequences/Complex/Sum Rule

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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 1

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:

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

$\blacksquare$


Proof 2

Let $\epsilon > 0$ be given.

Then $\dfrac \epsilon 2 > 0$.

We are given that:

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


Let:

$z_n = x_n + i y_n$
$w_n = r_n + i s_n$
$c = a + i b$
$d = l + i m$

where each of $x_n, y_n, r_n, s_n, a, b, l, m \in \R$ are real.


By definition:

$\sequence {z_n}$ converges to the limit $c = a + i b$ if and only if both:

$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \size {x_n - a} < \epsilon \text { and } \size {y_n - b} < \epsilon$

$\sequence {w_n}$ converges to the limit $d = l + i m$ if and only if both:

$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \size {r_n - l} < \epsilon \text { and } \size {s_n - m} < \epsilon$


where $\size x$ denotes the absolute value of $x \in \R$.


Then:

\(\ds \lim_{n \mathop \to \infty} \paren {z_n + w_n}\) \(=\) \(\ds \lim_{n \mathop \to \infty} \paren {x_n + i y_n + r_n + i s_n}\)
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \paren {x_n + r_n + i \paren {y_n + s_n} }\)
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \paren {x_n + r_n} + i \lim_{n \mathop \to \infty} \paren {y_n + s_n}\) Definition 1 of Convergent Sequence
\(\ds \) \(=\) \(\ds \paren {a + l} + i \paren {b + m}\)
\(\ds \) \(=\) \(\ds \paren {a + i b} + \paren {l + i m}\)
\(\ds \) \(=\) \(\ds c + d\)

$\blacksquare$


Also see


Sources

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