Sequence {an} that is eventually constant.

[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?

 

[matt grime]

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

 

[HallsofIvy]

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

 

[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.

 

[Nusc]

What about;

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?

 

[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

 

[HallsofIvy]

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).