Reciprocal Sequence is Strictly Decreasing/Proof 2

Theorem

The reciprocal sequence:

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

is strictly decreasing.

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

$\ds \frac 1 n - \frac 1 {n + 1}$

$=$

$\ds \frac {\paren {n + 1} - n} {n \paren {n + 1} }$

$\ds $

$=$

$\ds \frac 1 {n^2 + n}$

$\ds $

$>$

$\ds 0$

$\ds \leadsto \ \ $

$\ds \frac 1 n$

$>$

$\ds \frac 1 {n + 1}$

Hence the result, as $n$ was arbitrary.

$\blacksquare$