The following are three properties that may or may not belong to a sequence \((a_n)\) of strictly positive real numbers: \begin{align} (P_1) \quad & a_n \to 0 \text{ as } n \to \infty.\\ (P_2) \quad & \sum_{n=1}^\infty a_n \text{ converges}.\\ (P_3) \quad & \text{There is a constant } C \text{ such that } na_n < C \text{ for all values of } n. \end{align} For each pair of integers \((i,j)\) with \(1 \leq i \leq 3\), \(1 \leq j \leq 3\), \(i \neq j\), establish whether the statement `\(P_i\) implies \(P_j\)' is true or false. [You may quote without proof the behaviour of standard series.]
Explain carefully what is meant by the statement that a function of a real variable \(x\) is continuous at a particular value \(x_0\) of \(x\). The function \(f(x)\) of the real variable \(x\) takes the value 0 whenever \(x\) is irrational, and the value \(x^2(1-x^2)\) whenever \(x\) is rational. Find all values of \(x\) at which \(f(x)\) is continuous. Determine which (if either) of the following statements defines a real number, and find each number so defined: