I may have done it, must go to bed now:
The ##\alpha## is the root of an ##n-##th order polynomial with integer coefficients:
\begin{align*}
f (x) = \sum_{i=0}^n a_i x^i
\end{align*}
Put
\begin{align*}
\tilde{\alpha} = \frac{p}{q} .
\end{align*}
If ##f (x)## has rational roots then denote them as ##\{ r_1 , \dots , r_k \}##.
Case (a) ##\tilde{\alpha} \not\in \{ r_1 , \dots , r_k \}## and ##|\alpha - \tilde{\alpha}| \leq 1##.
We have the identity
\begin{align*}
f (x) & = \sum_{i=1}^n a_i x^i - \sum_{i=1}^n a_i a^i
\nonumber \\
& = \sum_{i=1}^n a_i (x - a) (x^{i-1} + x^{i-2} a + \cdots + a^{i-1})
\nonumber \\
& = (x - a) \sum_{i=1}^n a_i \sum_{j=0}^{i-1} x^{i-1-j} a^j
\end{align*}
Application of the triangle inequality ##|\tilde{\alpha}| = |\tilde{\alpha} - \alpha + \alpha| \leq |\alpha - \tilde{\alpha}| + |\alpha|## together with ##|\alpha - \tilde{\alpha}| \leq 1## implies ##|\tilde{\alpha}| \leq 1 + |\alpha|##. We use this to obtain the inequality:
\begin{align*}
|f (\alpha) - f (\tilde{\alpha})| & \leq |\alpha - \tilde{\alpha}| \sum_{i=1}^n |a_i| \sum_{j=0}^{i-1} |\alpha^{i-1-j} \tilde{\alpha}^j |
\nonumber \\
& = |\alpha - \tilde{\alpha}| \sum_{i=1}^n |a_i| \sum_{j=0}^{i-1} |\alpha^{i-1}| | \alpha^{-j} (1 + |\alpha|)^j |
\nonumber \\
& = |\alpha - \tilde{\alpha}| \sum_{i=1}^n |a_i \alpha^{i-1}| \sum_{j=0}^{i-1} \left( 1 + \frac{1}{|\alpha|} \right)^j
\nonumber \\
& = |\alpha - \tilde{\alpha}| \sum_{i=1}^n |a_i \alpha^{i-1}| \frac{\left( 1 + \frac{1}{|\alpha|} \right)^i - 1}{\left( 1 + \frac{1}{|\alpha|} \right) - 1}
\nonumber \\
& = |\alpha - \tilde{\alpha}| \sum_{i=1}^n |a_i| \left( (|\alpha| + 1)^i - |\alpha|^i \right)
\nonumber \\
& = |\alpha - \tilde{\alpha}| C_\alpha
\end{align*}
where ##C_\alpha = \sum_{i=1}^n |a_i| \left( (|\alpha| + 1)^i - |\alpha|^i \right) > 0##. We have
\begin{align*}
|f ( \tilde{\alpha} )| & = \left|\frac{a_0 q^n + a_1 q^{n-1} p + \cdots + a_n p^n}{q^n} \right| \geq \frac{1}{q^n}
\end{align*}
as the numerator is a nonzero integer, and so
\begin{align*}
|\alpha - \tilde{\alpha}| & \geq \frac{1}{C_\alpha} |f ( \tilde{\alpha} )| \geq \frac{1}{C_\alpha} \frac{1}{q^n}
\end{align*}
Case (b) ##\tilde{\alpha} \not\in \{ r_1 , \dots , r_k \}## and ##|\alpha - \tilde{\alpha}| > 1##.
\begin{align*}
|\alpha - \tilde{\alpha}| & > 1 \geq \frac{1}{q^n}
\end{align*}
Case (c) ##\alpha \not\in \{ r_1 , \dots , r_k \}##, ##\tilde{\alpha} \in \{ r_1 , \dots , r_k \}##.
Choose ##C_r = \min_i | \alpha - r_i| > 0##. Then
\begin{align*}
|\alpha - \tilde{\alpha}| & \geq C_r \geq \frac{C_r}{q^n}
\end{align*}
(If there are no rational roots, ignore case (c)).
Case (d) Say ##\alpha \in \{ r_1 , \dots , r_k \}## and ##\tilde{\alpha} \in \{ r_1 , \dots , r_k \}## (in which case ##k > 1##). Set ##\overline{C}_r = \min_{i \not= j}|r_i - r_j| > 0##,
\begin{align*}
|\alpha - \tilde{\alpha}| & \geq \overline{C}_r \geq \overline{C}_r \frac{1}{q^n}
\end{align*}
(If there are no rational roots, ignore case (d)).
Finally, we can write
\begin{align*}
c = \min \left\{
\begin{matrix}
\frac{1}{C_\alpha} & : |\alpha - \frac{p}{q}| \leq 1, \quad \frac{p}{q} \not\in \{ r_1 , \dots , r_k \} \\
1 & : |\alpha - \frac{p}{q}| > 1, \quad \frac{p}{q} \not\in \{ r_1 , \dots , r_k \} \\
C_r & : \alpha \not\in \{ r_1 , \dots , r_k \} , \quad \frac{p}{q} \in \{ r_1 , \dots , r_k \} \\
\overline{C}_r & : \alpha , \frac{p}{q} \in \{ r_1 , \dots , r_k \}
\end{matrix}
\right.
\end{align*}
then
\begin{align*}
\left| \alpha - \frac{p}{q} \right| & \geq \frac{c}{q^n}
\end{align*}
where ##c > 0##.