The nontrivial direction is a special case of the Lemma below, with $\rm\:D =\:$ integral closure of $\rm\:R.\:$
Lemma $\ $ Suppose that $\rm\:D = \mathbb Z\:$ (or any Noetherian integrally closed domain, e.g. any PID), and suppose that $\rm\:w\:$ is a fraction over $\rm\:D\:$ such that some unbounded sequence of powers of $\rm\:w\:$ has a common denominator $\rm\:0 \ne d\in D,\:$ i.e. $\rm\:d\!\:w^{n_i}\in D\:$ for all $\rm\:n_i.\:$ Then $\rm\:w\in D.$
Proof $\ $ By ACC the sequence of ideals $\rm (d, dw^{n_1}, dw^{n_2},\ldots)$ eventually stabilizes, which implies that for some $\rm\:k\:$ we have $\rm\: dw^{n_k}\in (dw^{n_{k-1}},\ldots, dw^{n_1}, d),\:$ which implies
$\rm d\: w^{n_k} + c_{n_{k-1}} d\: w^{n_{k-1}} +\:\! \cdots +\: c_{n_1} d\: w^{n_1} + d\: =\: 0$
Cancelling $\rm\:d\:$ yields $\rm\:w\:$ is integral over $\rm\:D,\:$ hence $\rm\:w\in D,\:$ since $\rm\:D\:$ is integrally closed. $\ $ QED
Remark $\ $ Elements whose powers have such a common denominator are called almost integral. It is clear that integral elements are almost integral. The above shows that the converse holds true in Noetherian domains. This is employed implicitly in Dedekind's work on ideal theory.
Here is a typical application from my answer to this prior question.
Show $\rm\ a\mid b^2,\: b^3\mid a^4,\: a^5\mid b^6,\: b^7\mid a^8 \:\cdots\:\Rightarrow\: a = b\:$ for $\rm\:a,b\in\mathbb Z_+$
Hint $\rm\ \ \forall\: n\in\mathbb N:\ \ a\:\!\left(\dfrac{a}b\right)^{4n+3}\!\in\mathbb Z,\:\ b\:\!\left(\dfrac{b}a\right)^{4n+1}\!\in \mathbb Z\ \ \Rightarrow\ \dfrac{a}b,\:\dfrac{b}a\in\mathbb Z\ \ \Rightarrow\ \ a = \pm b\ \ \ $ QED