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.

Proof

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.

$\blacksquare$