\(a_0, a_1, a_2, \ldots\) is a sequence of real numbers. Explain carefully what the following statements mean: \begin{align} (i) \quad & a_n \to a \text{ as } n \to \infty;\\ (ii) \quad & \sum_{n=0}^{\infty} a_n = b. \end{align} Show from first principles that if \(|x| < 1\) then \(x^n \to 0\) as \(n \to \infty\), and $$\sum_{n=0}^{\infty} x^n = (1-x)^{-1}.$$ [You may find the substitution \(|x| = 1/(1+y)\) helpful.]
The function \(f(x)\) is defined on the interval \(0 < x < 1\) as follows: (a) if \(x\) is rational, and \(x = p/q\) in lowest terms, then \(f(x) = 1/q\); (b) if \(x\) is irrational, \(f(x) = 0\). Show that if \(\epsilon\) is greater than \(0\), there are only finitely many points of the interval for which \(f(x) \geq \epsilon\). Deduce that \(f\) is continuous at the irrational points of the interval, and is discontinuous at the rational points.