Big O and equality

Question

On this problem, I'm not sure what Big O definition they are referring. How would the big o definition help show this? I get that $|y-y_h|$< $M\beta (h)$, where $y_h$ is an approximate value to y at some point h, but don't really understand the question

Use the definition of $O$ to show that if $y = y_h + O(h^p)$, then $hy = hy_h + O(h^{p+1})$.

Answer 1

I think that doesn't need too much work on because we know that:

$$f\cdot O(g)=O(fg)\quad\quad\quad (1) $$ Thus, if we multiply the first equation by $h$. Then, we use $(1)$ to find the wanted answer.