# Help proving/verifying a proof of a theorem

I have been trying to prove this theorem which I think is wrong. Theorem: Assume that $$f(h)=p(h)+O(h^n),g(h)=q(h)+O(h^m)$$and $$r=min\{m,n| m,n \in \mathbb{Z}_+\}$$ then $$f(h)+g(h)=p(h)+q(h)+O(h^r)$$ My views are,if we suppose that WLOG $$m\leq n$$ then $$h^m=O(h^n)$$ therefore $$O(h^m)+O(h^n)=O(h^n)$$ contrary to the proposition. So my question; am I right?if not where am I going wrong and how can I then prove the theorem above?

## 1 Answer

In principle you are right. But there is one detail missing: $$h→0 \qquad \text{or}\qquad h→∞?$$

Depending on that, your assumption is either true or false.

• $h→0$: It holds $h^m=\mathcal{O}(h^n)$ for $m≥n$.
• $h→∞$: It holds $h^m=\mathcal{O}(h^n)$ for $m≤n$.

And the proof of this is straight forward:

$$\frac{h^m}{h^n}=h^{m-n}→ … \tag{*}$$

The limit now depends on $m-n≥0$ or $m-n≤0$, and on $h→0/∞$.
If $(*)$ tends to a value $a<∞$, then it holds $h^m=\mathcal{O}(h^n)$.

Edit: The reason why I mention this is, that in numerics usually $h$ stands for a small value. Examples are the size of a grid cell in FEM/FDM, or the step-size in numerical methods for ODEs, or a small increment in a difference quotient.

• I get it now in this case I was wrong as $h\rightarrow 0$ – Tom Carter Oct 16 '17 at 7:40