Homework Help: Bounded sequence

1. Feb 21, 2010

jeff1evesque

1. The problem statement, all variables and given/known data
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.

2. Relevant 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.

3. 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 arguement (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?

2. Feb 21, 2010

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?

Last edited: Feb 21, 2010