Combination Theorem for Sequences/Real/Quotient Rule/Corollary

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Theorem

Let $\sequence {x_n}$ be a sequence in $\R$.


Let $\sequence {x_n}$ be convergent to the following limit:

$\displaystyle \lim_{n \mathop \to \infty} x_n = l$

Then:

$\displaystyle \lim_{n \mathop \to \infty} \frac 1 {x_n} = \frac 1 l$

provided that $l \ne 0$.


Proof

Follows directly from Quotient Rule for Real Sequences, setting

$\sequence {y_n} := \sequence {x_n}$

and:

$\sequence {x_n} := \tuple {1, 1, 1, \ldots}$

$\blacksquare$


Sources