Big o notation on both sides Can someone explain what exactly means $o(o(f(x))) = o(f(x))$?
I do understand what little oh means by definition(fraction limit of two functions converges to zero) , but somehow i don't get the idea of Big o or little o on both sides.
 A: I understand this expression in the following way: "any function $h(x)$ which is infinitely small compared to some other function $g(x)$, which, in turn, is infinitely small compared to function $f(x)$ (interpretation of left part of expression ends here) is itself infinitely small compared to $f(x)$".
Note, that "=" is not usual equality sign here, if you swap the left and right parts of this expression, it will not be correct any more.
UPDATE.
Well, in this particular example it IS possible to swap left and right parts of expression and it still will be correct. (Thanks to J.G. - author of another answer to this question).
In this example it is possible to interpret the expression as "set of functions described by the left part and the set of functions described by the right side are equal".
But I am quite sure I encountered expressions like $o(x^2) = o(x)$ meaning "our function is infinitesimal compared to $x^2$, so it is also infinitesimal compared to $x$"
A: Since $o(f)$ is a set of functions, $o(o(f))$ is an abuse of notation for $\bigcup_{g\in o(f)}o(g)$. In particular, you can prove $h\in o(f) \iff\exists g\in o(f)(h\in o(g))$ using this definition. In fact, you can choose $g=\sqrt{|fh|}$.
