# Reciprocal Sequence is Strictly Decreasing

## Theorem

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

## Proof 1

$\blacksquare$

## 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.

$\blacksquare$