Think, for example, of the intermediate value theorem, which asserts the existence of a point [xi], in the [a,b] interval of the horizontal axis, where a continuous function f, such that f(a) > 0, and f(b) < 0, vanishes, that is f([xi]) = 0.

Consider any i < j such that [w.sub.i] and [w.sub.j] are different kinds of parentheses, either [x.sub.i] < f([x.sub.i]) and f([x.sub.j]) < [x.sub.j] or f([x.sub.i]) < [x.sub.i] and [x.sub.j] < f([x.sub.j]) thus by the intermediate value theorem f admits a fixed point in the interval [[x.sub.i], [x.sub.j]], i.e., there is a * somewhere in between every pair of different parentheses in w(f), or equivalently the word w(f) cannot contain the factor [] nor the factor ][.

By the intermediate value theorem, there is an angle [theta] [member of][[[theta].sub.1], [theta].sub.2]] such that [[omega].sub.T]([theta]) = 1[square root of (3)].

Now in an updated and expanded third edition, "A Concise Introduction To Pure Mathematics" by Martin Liebeck provides an informed and informative presentation into a representative selection of fundamental ideas in mathematics including the theory of solving cubic equations, the use of Euler's formula to study the five Platonic solids, the use of prime numbers to encode and decode secret information, the theory of how to compare the sizes of two infinite sets, the limits of sequences and continuous functions, the use of the intermediate value theorem to prove the existence of nth roots, and so much more.

The essential ingredients of the proof are properties of the determinant of the matrix which describes the interpolation problem and the intermediate value theorem for real valued, continuous functions.