= Solution
Suppose $\chi_1$ and $\chi_2$ are two bound-state solutions with the same energy. Their <Wronskian>
$$
W=\chi_1\chi_2'-\chi_1'\chi_2
$$
satisfies $W'=0$ because the <Schrödinger equation> contains no first-derivative term. Hence $W$ is constant. Both <wavefunctions> and their <derivatives> decay at infinity, so $W\to0$ there and therefore $W\equiv0$. The two solutions are linearly dependent. Thus every one-dimensional bound-state energy is <nondegeneracy of one-dimensional bound states>[nondegenerate].
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