Real analysis question on subsequence

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February 13th 2013, 03:03 PM
sakuraxkisu

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 :)
February 13th 2013, 03:37 PM
Plato

Re: Real analysis question on subsequence

Quote:

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 $\left( {a_n } \right)\not \to L$ means that
$\left( {\exists \varepsilon > 0} \right)\left( {\forall N} \right)\left( {\exists n_N > N} \right)\left[ {\left| {L - a_{N_n } } \right| \geqslant \varepsilon } \right]$ .

Find the first $N_1$ then apply the above and find $N_2>N_1$ that works.

By induction, find a sequence of integers $N_1<N_2<\cdots<N_m<\cdots.$
February 16th 2013, 12:41 AM
SeirraFalcom

Re: Real analysis question on subsequence

Plato, great explanation, solved my other problem in the series also. I am glad that I bumped into this thread.
March 9th 2013, 02:25 AM
SeirraFalcom

Re: Real analysis question on subsequence

Quote:

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.