# Real Sequence/Examples/n over (n+1)

## Examples of Real Sequence

The real sequence $S$ whose first few terms are:

$\dfrac 1 2, \dfrac 2 3, \dfrac 3 4, \dotsc$

can be defined by the formula:

$S = \sequence {\dfrac n {n + 1} }_{n \mathop \ge 1}$

$S$ is strictly increasing.

## Proof

Let $s_n$ denote the $n$th term of $S$.

We have:

 $\ds s_{n + 1} - s_n$ $=$ $\ds \dfrac {n + 1} {n + 2} - \dfrac n {n + 1}$ $\ds$ $=$ $\ds \paren {1 - \dfrac 1 {n + 2} } - \paren {1 - \dfrac 1 {n + 1} }$ $\ds$ $=$ $\ds \dfrac 1 {n + 1} - \dfrac 1 {n + 2}$ $\ds$ $=$ $\ds \dfrac {\paren {n + 2} - \paren {n + 1} } {\paren {n + 1} \paren {n + 2} }$ $\ds$ $=$ $\ds \dfrac 1 {\paren {n + 1} \paren {n + 2} }$ $\ds$ $>$ $\ds 0$

Hence $S$ is increasing by definition.

$\blacksquare$