Hello again. I'm stuck on how to word my proof exactly or put it in notation or to find out if what I'm writing passes as proof. I'm pretty sure that my proof is right only I think I'm looking for guidelines on how to present it so that it's more readable and easier to understand.

Could you guys critique my proof writing?

Here is the problem:

Show that

for all n.

Showing the upper bound is easy because for beyond

there is always a negative fraction with more magnitude for every positive fraction and in the case that

is odd the left fraction is not enough to make up the gap to push the sum above 1.

To prove the case for the lower term assume again first that

is even. There are then couplets of terms after

of a greater positive fraction and a lesser negative fraction. That progression can only increase after

therefore it is bounded below. In the case that n is odd there will simply be a positive fraction added therefore it does not affect our proof of a lower bound.