Borel lemma : wikipedia proof

Question

In the proof of Borel's lemma, I don't understand why we use $\psi\left(\frac{t}{\epsilon_m}\right)$ for a sufficient small $\epsilon_m$ and not $\psi(t\cdot \epsilon_m)$, as you need to keep derivatives small enough to ensure convergence.

Answer 1

You need to have small supports of $t\mapsto \psi(t/\varepsilon_m)$. Of course, the derivatives become large and you have to deal with this in the proof of Borel's lemma.