Reciprocal Sequence is Strictly Decreasing

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Theorem

The reciprocal sequence:

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

is strictly decreasing.


Proof 1

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

$\blacksquare$


Proof 2

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

\(\displaystyle \frac 1 n - \frac 1 {n + 1}\) \(=\) \(\displaystyle \frac{\left({n+1}\right)-n}{n\left({n + 1}\right)}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {n^2 + n}\) $\quad$ $\quad$
\(\displaystyle \) \(>\) \(\displaystyle 0\) $\quad$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle \frac 1 n\) \(>\) \(\displaystyle \frac 1 {n+1}\) $\quad$ $\quad$

Hence the result, as $n$ was arbitrary.

$\blacksquare$


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