The exercise 9 section 7 chapter 1 in Guillemin & Pollak state the next
Prove directly Morse lemma for real line $\mathbb{R}$.(Hint:Use this elementary calculus lemma: for any function on $\mathbb{R}$ and any point $a\in\mathbb{R}$ there is another function $g$ such that $$f(x)=f(a)+(x-a)f'(a)+(x-a)^2g(x)$$ I do not quite understand is that I have to do, if someone have any idea please share. My idea is the next:
I'll do everything around zero, then $$f(x)=f(0)+x\cdot f'(0)+x^2\cdot \frac{f''(0)}{2}+o(x^3)$$ this for taylor theorem. Since we suppose $0$ singular nondegerate $f'(0)=0$, thus $$f(x)=f(0)+x^2\cdot \frac{f''(0)}{2}+o(x^3)=f(0)+\frac{f''(0)}{2}\cdot x^2(1+o(x))=f(0)+\frac{f''(0)}{2}\cdot (x\sqrt{1+o(x)}) ^2$$ Then for $y=x\sqrt{1+o(x)}$ I have that $$\frac{dy}{dx}\mid_{x=0}\not=0$$ And for the implicit function theorem $x=x(y)$ then $$f(x(y))=\bar{f}(y)=f(0)+\frac{f''(0)}{2}y^2$$ It is right? Should be easier with the hint?
A minor comment on your use of little-o notation: You should drop the power of $x$ by one throughout. The statement $f = o(g)$ is the statement that $f/g \rightarrow 0$.
Yes. The hint is a version of Taylor's theorem, which you use in a slightly different form right at the beginning. Your $C(1+o(1))$ is their $g(x)$.
As long as you know why a function that's $o(x^2)$ is also $x^2$ times a function that's $o(1)$, your proof is as thorough as they were looking for.