= Solution
Let $x\in\operatorname{Frac}(D)$ satisfy a <monic polynomial> equation
$$
x^n+a_{n-1}x^{n-1}+\cdots+a_0=0,\qquad a_i\in D.
$$
If $x=0$, there is nothing to prove. Suppose $x\notin D$. The defining property of a <valuation ring> gives $y=x^{-1}\in D$. Multiplying the equation by $y^{n-1}$ yields
$$
x=-a_{n-1}-a_{n-2}y-\cdots-a_0y^{n-1}\in D,
$$
a contradiction. Therefore
$$
\boxed{D\text{ is an integrally closed domain}.}
$$
This proof uses only the comparison of an element with its inverse; no discreteness or finite generation of the <valuation> is needed.
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