I don't think I gave this problem before.

Let be an injection. Show that

with equality only if is the embedding of in .

Printable View

- May 23rd 2010, 11:15 PMBruno J.Harmonic numbers
I don't think I gave this problem before.

Let be an injection. Show that

with equality only if is the embedding of in . - May 27th 2010, 02:12 AMLaurent
I guess you mean , not ?

__Spoiler__: - May 27th 2010, 09:02 AMBruno J.
Haha, yes! I was just having this discussion with friends a few days ago; they claimed that unambiguously meant the set of positive integers, while I claimed that was sometimes included... I guess you can never be too precise.

Nice solution! It's completely different from mine, I'll post it a bit later. - May 31st 2010, 10:43 AMBruno J.
Alright, sorry for the delay; here's my solution.

Suppose is such that the sum on the right is minimal. Then we can assume to be a permutation of ; indeed, order the values in increasing order and replace them respectively by ; we minimize the sum in such a way. Now suppose there exist with and ; suppose we switch the values of in the sum. The difference between the two sums is , meaning we have minimized the sum some more, contradicting the hypothesis. Therefore .