# Sequence {an} that is eventually constant.

1. Nov 27, 2005

### Nusc

The Sequence {an} is eventually constant.

A sequence {an} is eventually constant if there exists an index n*EN s.t. an = C for all n>n*.

What am I missing?

2. Nov 27, 2005

### matt grime

We have no idea, what do you think you're missing?

3. Nov 27, 2005

### HallsofIvy

Staff Emeritus
In other words, the sequence
1, 6, 9, 8.9, -33, 51.6, -23, 7, 7, 7, 7, ....7, 7, 7....
is "eventually constant".

4. Nov 27, 2005

### benorin

Your definition seems ok, that is I googled for it a found that if a sequence is eventually constant, then its range is a finite point set, which is consistant with your definition.

5. Nov 27, 2005

### Nusc

The sequence {an} is eventually increasing.

A sequence {an} is eventually increasing if and only if there exists an index n*E N s.t. an _<_ am for all m>n _>_ n*

Is that right?

6. Nov 28, 2005

### matt grime

I've never seen anyone need or bother to define that concept before, but if they were to then that is what i imagine it would look like

7. Nov 28, 2005

### HallsofIvy

Staff Emeritus
It is generally true that any finite number of terms in a sequence can be changed or eliminated without changing the limit of the sequence. That is why "eventually" some property is important. A sequence that is "eventually constant" can be treated like a constant sequence (so its limit is that constant) and a sequence that is "eventually increasing" can be treated like an increasing sequence (so if it has an upper bound it converges).