Homework Help: Convergence of sequences

1. May 5, 2007

pivoxa15

1. The problem statement, all variables and given/known data
Does
A subsequence of a sequence X converges to a point in I => The sequence X in I converges to a point in I
?

3. The attempt at a solution
I think yes because the subsequence is the sequence itself minus a few finite number of points. Since they both are in the same set I, I can't see why not.

Last edited: May 5, 2007
2. May 5, 2007

Office_Shredder

Staff Emeritus
A subsequence can be lacking an infinite number of points in I. If a sequence is 1,-1,1,-1,1,-1,...

the subsequence

1,1,1,1,1...

certainly converges. You can tell me what you think about the statement

3. May 5, 2007

pivoxa15

Good example.

The question should be
Does
A subsequence of a sequence X converges to a point in I <= The sequence X in I converges to a point in I
?

Now it should be yes.

But we genearlly refer to seq and subseq as containing an infinite number of points.

4. May 5, 2007

quasar987

If $$f:\mathbb{N}\rightarrow I$$ is a sequence in a set I, then a subsequence of f is a sequence of the form h = f o g, where $$g:\mathbb{N}\rightarrow\mathbb{N}$$ is a strictly increasing sequence of natural numbers.

I like to think of g as a discriminating function that picks which guys from f it wants in its kickball team.. or which girls does the Maharajah wants in its harem, or... any such pictorial analogy to remember the definition.

Last edited: May 5, 2007
5. May 5, 2007

quasar987

And it goes farther too: The sequence X in I converges to a point y in I <=> Every subsequence of X converges to y.

6. May 6, 2007

pivoxa15

It a little subtle. Every is essential. I didn't have every in my original statement so no if and only if condition.