Combination Theorem for Sequences/Product Rule
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Theorem
Product Rule for Real Sequences
Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $\R$.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent to the following limits:
- $\ds \lim_{n \mathop \to \infty} x_n = l$
- $\ds \lim_{n \mathop \to \infty} y_n = m$
Then:
- $\ds \lim_{n \mathop \to \infty} \paren {x_n y_n} = l m$
Product Rule for Complex Sequences
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:
- $\ds \lim_{n \mathop \to \infty} \paren {z_n w_n} = c d$