We have that ${a_n}_{nin N}$ is a succession of real numbers that is NOT bounded below.

(1) Must all the elements of $a_n$ be negative?

(2) If there are positive elements, those must be finite?

(3) Must there be INFINITE negative elements?

To answer 1 and 2 I find a succession that is not bounded below, that has infinite positive elements:

$${a_n}_{nin N}={-1,2,-3,4,-5,6,…}$$

$$a_n=n(-1)^n$$

Which is not bounded (below), and has infinite positive and negative elements.

Now, the third question **I’m not sure how to prove that there MUST be infinite negative elements on a succession which is not bounded below.**

What I thought of until now:

${a_n}_{nin N}$ Is not bounded below, then, by definition:

$nexists min R , forall nin N : a_n geq m$

This means that there will always be an item on the succession that is smaller than a previous item:

$forall nin N , exists kin N : a_{n+k}leq a_n$

And as $N$ is not bounded, the items smaller than the previous one are infinite?