Quotient Rule for Real Sequences/Corollary

Corollary to Quotient Rule for Real Sequences

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

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

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

Then:

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

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

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.2$: Real Sequences: Proposition $1.2.11 \ \text {(c)}$