Liouville approximation theorem

If an irrational algebraic number has degree $d$, then $|\alpha -p/q|>C(\alpha )q^{-d}$. Approximations of arbitrarily larger polynomial order therefore prove that a Liouville number is transcendental.

For the degree-$d$ minimal polynomial $P\in \mathbb{Z}[X]$ of $\alpha$, the nonzero integer $q^{d}P(p/q)$ gives $|P(p/q)|\ge q^{-d}$. On a fixed neighbourhood of $\alpha$, the mean value theorem boundsOutside that neighbourhood the desired lower bound is immediate, giving $|\alpha -p/q|>c(\alpha )q^{-d}$ after decreasing the constant.

For distinct reduced fractions $p/q$ and $u/v$,If $q>v$, this is strictly greater than $q^{-2}$.

Every convergent series of positive rational terms has a subseries whose rational partial sums $p_j/q_j$ satisfyChoose each new term small enough to enforce all previous tail bounds. Separation of reduced fractions first makes $\alpha$ irrational, and the Liouville approximation theorem then rules out every finite algebraic degree.