Thread: Real analysis question on subsequence

Hi, my question is:

Let L be a real number and let (a_n) be a sequence of real numbers that does not converge to L (that is, it is either divergent or its limit is not equal to L). Use the definition of convergence to L to show that for some e greater than 0, (a_n) has a subsequence (a_n(_k)) such that (a_n(_k)) isn't in the interval (L - e, L+e) for all natural numbers k.

Any help would be hugely appreciated

Originally Posted by sakuraxkisu

Let L be a real number and let (a_n) be a sequence of real numbers that does not converge to L (that is, it is either divergent or its limit is not equal to L). Use the definition of convergence to L to show that for some e greater than 0, (a_n) has a subsequence (a_n(_k)) such that (a_n(_k)) isn't in the interval (L - e, L+e) for all natural numbers k.

To say that means that
.

Find the first then apply the above and find that works.

By induction, find a sequence of integers

Plato, great explanation, solved my other problem in the series also. I am glad that I bumped into this thread.

Originally Posted by SeirraFalcom

Plato, great explanation, solved my other problem in the series also. I am glad that I bumped into this thread.

Update: I am stuck again in the test prep of my discrete math subject.