I am having trouble understanding both intuitively, and by example, statements involving big $O$ and small $o$ notation.
$1)$ $f(h) = O(h^p)$ then $ f(h) = O(h^q)$ for all $ q ≥ p$.
Why is the previous statement false? I thought that intuitively, any function that is bounded above by a power of $h$ will also be bounded above by a higher power of $h$.
$2)$ If $p < q, f(h) = o(h^p)$ and $ g(h) = O(h^q)$, then $ f(h) + g(h) = o(h^p)$.
Why is the previous statement true? How can we relation small $o$ to big $O$?
$3)$ If $f(h) = o(h^p)$ and $ g(h) = O(h^q)$, then $f(h)g(h) = o(h^{p+q})$.
Why is the previous statement true? I have the same problem with understanding this statement since I don't see how to manipulate expressions with both big $O$ and small $o$.
EDIT: this question refers to $h\rightarrow 0$