# Sequence and subsequence - real analysis

1. Feb 22, 2015

### Dassinia

Hello,
Solving last exam and stuck in this exercise
1. The problem statement, all variables and given/known data
Consider an increasing sequence {xn} . We suppose ∃ x∈ℝ and {xnk} a sebsequence of {xn} and xnk→x
a/ Show that for any n∈ℕ , ∃ k∈ℕ as n≤nk
b/ Show that xn→x
2. Relevant equations
3. The attempt at a solution

For b/ it is easy.
But for a/ I really don't know how to do that

thanks

2. Feb 23, 2015

### Stephen Tashi

I don't know how to interpret that statement. What is it that happens "as" $n \le n_k$ ?

3. Feb 23, 2015

### HallsofIvy

If you mean "n∈ℕ , ∃ k∈ℕ such that n≤nk" that just says that, given any integer n, there exist an "nk", an index from the subsequence, larger than n. And that comes from the fact that the subsequence is infinite.

4. Feb 23, 2015

### Dassinia

Yes sorry it is such that , it was late !
I dont know where to start from to get to this result ?

5. Feb 25, 2015

### SammyS

Staff Emeritus
Well, what do you mean by nk ?

Isn't {nk} an increasing sequence in , so that {xnk} is a subsequence ?