Combination Theorem for Sequences/Real
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Theorem
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:
- $\ds \lim_{n \mathop \to \infty} x_n = l$
- $\ds \lim_{n \mathop \to \infty} y_n = m$
Let $\lambda, \mu \in \R$.
Then the following results hold:
Sum Rule
- $\ds \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$
Difference Rule
- $\ds \lim_{n \mathop \to \infty} \paren {x_n - y_n} = l - m$
Multiple Rule
- $\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$
Combined Sum Rule
- $\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$
Product Rule
- $\ds \lim_{n \mathop \to \infty} \paren {x_n y_n} = l m$
Quotient Rule
- $\ds \lim_{n \mathop \to \infty} \frac {x_n} {y_n} = \frac l m$
provided that $m \ne 0$.