Reciprocal Sequence is Strictly Decreasing
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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 $n \in \N_{>0}$.
\(\ds \frac 1 n - \frac 1 {n + 1}\) | \(=\) | \(\ds \frac {\paren {n + 1} - n} {n \paren {n + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {n^2 + n}\) | ||||||||||||
\(\ds \) | \(>\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac 1 n\) | \(>\) | \(\ds \frac 1 {n + 1}\) |
Hence the result, as $n$ was arbitrary.
$\blacksquare$