Prove: x_m Is Not Bounded Above, x_m Does Not Converge

[jeff1evesque]

Homework Statement

Let x_m = 1 + \frac{1}{2} + \frac{1}{3} + ... \frac{1}{m}, m \in N.
Prove x_m is not bounded above and therefore x_m does not converge.

Homework Equations

We know from our class an important theorem stating that:
If sequence converges then the sequence is bounded.

Thus we can say if the sequence is not bounded then it is not convergent.

The Attempt at a Solution

By above (#2), i just have to show our sequence is not bounded. This means i have the following claim:
x_m is not bounded above if and only if given any S > 0 , there exists m such that x_m > S.

Question:
1. Do i have to prove both sides of the argument (if and only if)? Or can I just change my claim to a one sided (left to right)?

2. Can someone help me formulate some thoughts on how to begin this proof?

 

[Dick]

This is a lot like your other post. Here's a hint. 1/3+1/4>1/2. 1/5+1/6+1/7+1/8>1/2. 1/9+1/10+...+1/15+1/16>1/2. Why?