Grishka, nothing you have posted so far would have suggested that English wasn't your first language.

if this were serious rather than a "game", I would have been more cautious and meticulous).

Sorry, don't get that. Whether it's serious or a game, I can't see the point of doing it if you aren't going to do your best to get it right.

I'm sorry if we'v scared off the other readers but MtheGM must have known that he was setting a chalenge. If he doesn't like the answer, he has only himself to blame.

Maybe you or I should try and explain your friends proof in plain English. "10^n-1 ≤ x²" doesn't make a lot of immediate sense to someone who isn't used to mathematical notation.