Quotient Rule for Real Sequences/Corollary

Theorem
Let $X$ be one of the standard number fields $\Q, \R, \C$.

Let $\left \langle {x_n} \right \rangle$ be a sequences in $X$.

Let $\left \langle {x_n} \right \rangle$ be convergent to the following limit:
 * $\displaystyle \lim_{n \to \infty} x_n = l$

Then:


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

provided that $l \ne 0$ and $x_n \ne 0$ for $n = 1, 2, 3, \ldots$

Proof
Follows directly from Combination Theorem for Sequences/Quotient Rule, setting
 * $\left \langle {y_n} \right \rangle := \left \langle {x_n} \right \rangle$

and:
 * $\left \langle {x_n} \right \rangle := \left ({1, 1, 1, \ldots} \right)$