I like janvdl's answer. I would have never thought of that! I would have tried summing an infinite series, probably.

1)

Assume it is an integer:

1+ 1/2 + ... + 1/n = p/q
P and q have no common factors blah blah blah.

n + n/2 + ... + 1 = np/q
n(n-1) + n(n-1)/2 + ...+ 1 + (n-1) = n(n-1)p/q
n(n-1)(n-2) + n(n-1)(n-2)/2 + ...+ 1 + (n-2) + (n-1)(n-2) = n(n-1)(n-2)p/q

etc.

n! + (n-1)! = n!p/q (er- is this right?)
n + 1 = (n+1)p/q
i.f.f p=q, therefore assumption is wrong, blah blah blah?

Actually that's definitely wrong.