= Normalization forces Hermite-series termination
{c}
For the <Hermite differential equation> $f''-2\xi f'+(\epsilon-1)f=0$ with real $\epsilon$, the <wavefunction> $e^{-\xi^2/2}f$ is square integrable on the real line only if $f$ is a <polynomial>. To prove this without assuming that even and odd tails cannot cancel, project a square-integrable solution onto its even and odd parts first. Each projection remains square integrable and satisfies the same equation.
In either nonterminating parity sequence, $b_j=a_{2j+p}$ obeys
$$
\frac{b_{j+1}}{b_j}=\frac{4j+2p+1-\epsilon}{(2j+p+2)(2j+p+1)}\sim\frac1j.
$$
The tail has constant complex phase. For any fixed $0<\delta<1/2$, comparison with $(1-\delta)/(j+1)$ bounds its coefficient magnitudes below by $K(1-\delta)^j/j!$ for sufficiently large $j$. Thus on the positive real axis the tail dominates a positive multiple of $\xi^p e^{(1-\delta)\xi^2}$ after subtraction of finitely many terms. The <Gaussian function> factor leaves exponential growth $e^{(1/2-\delta)\xi^2}$, contradicting the <square-integrable> condition. Hence each nonzero parity sequence terminates. Its top coefficient forces $\epsilon=2n+1$, giving <Hermite oscillator quantization> and a <Hermite polynomial> of degree $n$.
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