# Reciprocal Sequence is Strictly Decreasing

## Contents

## Theorem

The reciprocal sequence:

- $\sequence {\operatorname {recip} }: \N_{>0} \to \R$: $n \mapsto \dfrac 1 n$

## Proof 1

Follows from Reciprocal Function is Strictly Decreasing and from Restriction of Monotone Function is Monotone.

$\blacksquare$

## Proof 2

Let $n \in \N_{>0}$.

\(\displaystyle \frac 1 n - \frac 1 {n + 1}\) | \(=\) | \(\displaystyle \frac {\paren {n + 1} - n} {n \paren {n + 1} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 {n^2 + n}\) | |||||||||||

\(\displaystyle \) | \(>\) | \(\displaystyle 0\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \frac 1 n\) | \(>\) | \(\displaystyle \frac 1 {n + 1}\) |

Hence the result, as $n$ was arbitrary.

$\blacksquare$