i've been stuck on this proof for hours:
suppose 0<p<q, a_n=a+O(n^(-p)) and b_n=b+O(n^(-q)) where O is the order of convergence as n approaches infinite. i need to prove that a_n*b_n=ab+O(n^(-p)), in another words that this function converges to ab with order of convergence n^(-p).
by definition, a sequence a_n converges to a with rate of convergence O(b_n) if there exists k such that abs (a_n-a)<=k*abs(b_n).
is it possible to prove (or even true) that abs (a_n*b_n-ab) <= abs (a_n-a)* abs (b_n-b)? this would allow us to find such a k.
thanks in advance for any help!