# Does a subsequence only have to have some terms

1. Feb 16, 2005

### eckiller

Does a subsequence only have to have "some" terms

This is an example from my text, which I do not understand.

Suppose (s_n)is a sequence of POSITIVE numbers such that inf{s_n | n in NaturalNumbers} = 0. The sequence need not converge or even be bounded , but it has a subsequence that converges monotonically.

They then give a recursive process on how to pick the elements of the subsequence. My problem is as follows. Suppose the sequence is:

s_n = (n - 10)^2

This is a shifted parabola (but sequence) that has an inf = 0. A subsequence must have infinite terms, correct? I can see how you can pick terms starting from the left hand side of the vertex, and pick a monotone decreasing terms up to the vertex (0, 0), but what about after that? s_n starts moving back up and goes to +infinity. So past a certain point, we can't pick anymore points off s_n for our subsequence and keep the monotonicity of it.

So what am I missing? Does a subsequence only have to have "some" terms from the main sequence it is a subsequence of?

2. Feb 16, 2005

### learningphysics

But this isn't a sequence of positive numbers s_10=0.

3. Feb 16, 2005

### HallsofIvy

Staff Emeritus

Do you see why that's important? If you allowed 0 in the sequence or if the inf was a positive number, then a sequence that included inf but all other numbers might be far away from it which would violate your conclusion.

The inf of the sequence is 0 but 0 itself cannot be in the sequence! Given any integer n, let ε= 1/n. Are there any members of the sequence less than ε?