(i) Give the complete details in the inductive argument in Part A of the proof of Theorem 1.35 above.
(ii) Give the details of the 'similar argument applied to the bn' which shows that /(c) < 0.
(Hi) We use various parts of Lemma 1.6 in our Theorem 1.35. Identify the points where we use Lemma 1.6.

Show that any real polynomial of odd degree has at least one root. Is the result true for polynomials of even degree? Give a proof or counterexample.

Every mid-summer day at six o'clock in the morning, the youngest monk from the monastery of Damt starts to climb the narrow path up Mount Dipmes. At six in the evening he reaches the small temple at the peak where he spends the night in meditation. At six о 'clock in the morning on the following day he starts downwards, arriving back at the monastery at six in the evening.

Suppose that we have a sequence x\, X2, of real numbers. Let [ao, bo] be any closed interval. Show that we can find a sequence of pairs of points an and bn such that