# Monotonicity of a sequence

1. Oct 28, 2005

### kreil

determine the monotonicity of [itex]a_n=\frac{n-1}{n}[/tex]

heres my work...

$$\frac{a_{n+1}}{a_n}=\frac{ \frac{n}{n+1}}{\frac{n-1}{n}}$$

$$=\frac{n^2}{n^2-1}>1$$

Therefore the sequence is monotone increasing.

But...when you look at the original expression for a_n, it looks like it is always LESS than one...does this impact anything at all?

Josh

2. Oct 28, 2005

### Tide

The sequence approaches the limit 1 from below so it's clearly monotonically increasing.

3. Oct 28, 2005

### quasar987

A sequence can be strictly increasing but never greater than a certain number. This is what limits are all about.

4. Oct 28, 2005

### HallsofIvy

It has an important impact! One of the fundamental properties of the real numbers is the "Monotone Convergence Property". If an increasing sequence of real numbers has an upper bound, then it converges.

What you have here is an increasing sequence that has every number larger than or equal to 1 as an upper bound. 1 is its "least upper bound" and so the sequence converges to 1.