Question
Let $c$ be constant. As $x \to 0$, $$ f = cx^2 + o(x^2). $$
Then, $$ f^2 = c^2 x^4 + o(x^4). $$
What I think
\begin{align*} f^2 &= c^2 x^4 + 2cx^2 o(x^2) + o(x^4) \\ &= c^2 x^4 + o(x) \cdot o(x^2) + o(x^4) \\ &= c^2 x^4 + o(x^3) + o(x^4) \\ &= c^2 x^4 + o(x^3).? \end{align*}
Why is it different?