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Starting with ${\theta }_0=\theta$, definewhile the denominator is nonzero. This gives the simple continued fractionA finite expansion is rational by evaluating it from the bottom. Conversely, for rational $\theta$, these steps are the Euclidean algorithm applied to numerator and denominator, so the remainders eventually vanish.

Define convergents byThe recurrence givesso induction yieldsIn particular,Since an irrational $\theta$ lies strictly between these convergents, the two approximation errors sum to this distance. If both displayed bounds in the question failed, their sum would be at leastby the arithmetic-geometric mean inequality, contradicting strict betweenness. Thus at least one bound holds.

The elementary symmetric functions areThe Fundamental theorem of symmetric polynomials says that every symmetric polynomial is uniquely a polynomial in $e_1,\dots ,e_n$.

If $r_1,\dots ,r_n$ are the roots of a monic polynomial, its discriminant isThis is symmetric in the roots, hence the theorem and Vieta's formulas express it polynomially in the coefficients. Equivalently,For $x^5+q$ this givesFor $f=x^5+p{x}^2+q$, a repeated root solves $f=f^{\prime }=0$. Elimination givesThus the discriminant vanishes exactly when $q=0$ or $108p^5+3125q^3=0$.