Combination Theorem for Sequences/Complex/Difference 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:

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


Then:

$\displaystyle \lim_{n \mathop \to \infty} \paren {z_n - w_n} = c - d$


Proof

From Sum Rule for Complex Sequences:

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

From Multiple Rule for Complex Sequences:

$\displaystyle \lim_{n \mathop \to \infty} \paren {-w_n} = -d$

Hence:

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

The result follows.

$\blacksquare$


Also see


Sources