Sequences and the sequences' arithmetics

I am asked if there is an arithmetic (The average of all elements of a sequence) with a limit of zero while all the elemts of the sequence are strictly positive and the sequence itself diverges to infinity. I would say no since for a given number M all elements that are bigger then some N_M are taller then this number M. Thus the arithmetic must deverge, too. However, this seems to easy and they would not ask like they do if there was no example. Any ideas? Thank you for help!