= Solution
Let $\lambda=mU_0a^2/\hbar^2\ll1$ and $ka\ll1$. Inside a region of size $a$, the <Schrödinger equation> estimates the curvature as $a^2\psi''=O(\lambda+(ka)^2)\psi$. Matching to the incoming and reflected waves constrains the slope as well. To leading order the interior wave has nearly constant value $C$, while its small accumulated <derivative> change encodes the potential's integrated strength.
This is not a Born assumption that $C$ is close to one. At very low $k$ the wave can be strongly reflected even by a weak potential, and $C$ can be small. The matching solution in the next part gives $C\sim k/(k+i m\gamma/\hbar^2)$. It implies a fractional variation across the range of order $ka+m|\gamma|a/\hbar^2$, which is small under the two specified conditions. Thus
$$
\boxed{\psi(x)=C[1+O(ka,\lambda)]\quad\text{within the potential range}.}
$$
The approximation requires a finite short-range potential with a controlled width and <integral>; it does not describe arbitrary long-range tails.
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