Convergence of Subsequences in a Set I

[pivoxa15]

Homework Statement

Does
A subsequence of a sequence X converges to a point in I => The sequence X in I converges to a point in I
?

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.

 

[Office_Shredder]

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

 

[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.

 

[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.

 

[quasar987]

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.

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

 

[pivoxa15]

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

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