Sequences and limits

Question

Let (a_n) be a sequence with a_n>0 for all natural numbers n. Assume that lim(a_n)=0. Show that the set of all numbers a_n has a maximum. That is, show that there is some number p, such that a_n <=a_p.

My idea: after a certain point all elements will tend toward 0 since lim(a_n)=0. At that point, take the maximum of the elements of (a_n) before that point. That is all I have so far. I need help with writing out the details...

You can also show that if there is no maximum, then $\limsup{x_n} \not=0$, providing a contradiction. — Euler....IS_ALIVE, Oct 20 '14 at 01:01

score 1 · Accepted Answer · answered Oct 20 '14 at 00:58

Convergence dictates that there is a value $N$ such that $\forall n > N \ : \ a_n < \frac{a_1}{2}$, i.e. we chose $\epsilon = \frac{a_1}{2}$. Now the set $\{a_n | n \leq N\}$ is finite, thus has a maximum which is obviously also the maximum of the entire sequence.

Sequences and limits

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