# Reciprocal of Null Sequence

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## Theorem

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

Let $\forall n \in \N: x_n > 0$.

Then:

- $x_n \to 0$ as $n \to \infty$ if and only if $\size {\dfrac 1 {x_n} } \to \infty$ as $n \to \infty$

### Corollary

- $x_n \to \infty$ as $n \to \infty$ if and only if $\size {\dfrac 1 {x_n} } \to 0$ as $n \to \infty$

## Proof

Suppose $x_n \to 0$ as $n \to \infty$.

Let $H > 0$.

So $H^{-1} > 0$.

Since $x_n \to 0$ as $n \to \infty$:

- $\exists N: \forall n > N: \size {x_n} < H^{-1}$

That is:

- $\size {\dfrac 1 {x_n} } > H$.

So:

- $\exists N: \forall n > N: \size {\dfrac 1 {x_n} } > H$

and thus:

- $\sequence {\size {\dfrac 1 {x_n} } }$ diverges to $+\infty$.

$\Box$

Suppose $\size {\dfrac 1 {x_n} } \to \infty$ as $n \to \infty$.

By reversing the argument above, we see that $x_n \to 0$ as $n \to \infty$.

$\blacksquare$

## Also known as

Some sources call this the **reciprocal rule**, but as that name is used throughout mathematical literature for several different concepts, its use is not recommended.

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 4$: Convergent Sequences: Exercise $\S 4.29 \ (4)$