Sequence is convergent if it has a convergent subsequence

[gottfried]

Homework Statement

Show that an increasing sequence is convergent if it has a convergent subsequence.

The Attempt at a Solution

Suppose x_jn is a subsequence of x_n and x_jn→x.

Therefore \existsN such that j_n>N implies |x_jn-x|<\epsilon
It follows that n>j_n>N implies |x_n-x|<\epsilon

Therefore x_n→x

The solution that I've been given is much more complicated I'm just wondering whether my simpler solution is also correct.

 

[jbunniii]

Homework Statement

Show that an increasing sequence is convergent if it has a convergent subsequence.

The Attempt at a Solution

Suppose x_jn is a subsequence of x_n and x_jn→x.

Therefore \existsN such that j_n>N implies |x_jn-x|<\epsilon
It follows that n>j_n>N implies |x_n-x|<\epsilon

It's true, but WHY does it follow? This is the key part of the proof, so you need to be explicit. You need to use the fact that x_n is an increasing sequence. This would not be true for an arbitrary sequence.

 

[gottfried]

The original question was an 'if and only if question' which means the reverse also had to be proved ie:

Show that if x_n → x that any subsequence of (x_n) also converges to x
My solution which is simliar to the answer given is

If x_n → x then given any  \epsilon> 0 there is an N such that n > N implies |x_n - x| < \epsilon . Now
consider a subsequence (x_jn). Then since jn \geq n > N we have that for any n > N,
|x_jn - x| < \epsilon  and so we conclude that the subsequence has the same limit

I believe this correct and this raises the question why does logic work in the one direction but no the other.

 

[jbunniii]

The original question was an 'if and only if question' which means the reverse also had to be proved ie:

Show that if x_n → x that any subsequence of (x_n) also converges to x
My solution which is simliar to the answer given is

If x_n → x then given any \epsilon> 0 there is an N such that n > N implies |x_n - x| < \epsilon . Now
consider a subsequence (x_jn). Then since jn \geq n > N we have that for any n > N,
|x_jn - x| < \epsilon and so we conclude that the subsequence has the same limit

I believe this correct and this raises the question why does logic work in the one direction but no the other.

It does work in both directions. If EVERY subsequence of x_n converges to x, then x_n converges to x. This is trivial, because x_n is a subsequence of itself.

But the hypothesis in the first part is weaker: x_n has *a* convergent subsequence. Without the additional assumption that x_n is increasing, this would not be enough to conclude that x_n converges.

A general sequence can have some subsequences which converge, and others which do not. For example, let x_n = 0 if n is even, and x_n = n if n is odd. The subsequence consisting of even indices converges, and the subsequence consisting of odd indices diverges. And of course the sequence itself does not converge.

 

[gottfried]

Thanks it makes sense.