It was not the "f(n) = n + f(10n)" that makes no sense, it was "f(1) approaches infinity". You want to show that f(n) approaches infinity as n goes to infinity. And, so, you do not want to show that there is no number p such that f(1)< p, you want to show that there is no number p such that f(n)< p for **all**.

Given any integer, p, look at f(p)= p+ f(10p). As long as you can show that f(n) is always positive, that is larger than p.

But even that would NOT be enough to show that f(n) goes to infinity- you would also have to show that f(n) is "eventually increasing"- that is, that for some N, if n> N, then f(n+1)> f(n).