x, and ak E Q for all k. Thus we have a sequence of rational numbers converging to x in this case as well. ) • Remark. Because lR is complete and because the set Q c lR is dense in IR, it follows that any set of numbers that contains limits for all its Cauchy sequences and that contains Q must also contain JR.

This contradiction proves the Heine-Bore) theorem. 76 Show that a closed finite interval [a,b] is not an open set. 77 Show that a half-closed finite interval (a, b) is not an open set. , and for each x E 0 Jet rx be defined as in the proof of Theorem 1. 1. 79 31 The empty set 0 satisfies the definition of being open. Explain. 80 Find an open cover of the interval ( -1, 1) that has no finite subcover. Justify your claims. 81 Find an open cover of the interval ( -oo, oo) that has no finite subcover.

Find sup(Tn) and inf(T11 ), where Tn is the nth tail of the sequence, and explain. Find lim inf x 11 and lim sup x ... 36 (xn Prove or give a counterexample: If Xn increases and Yn increases, then + Yn) is monotone. 37 Prove or give a counterexample: if Xn increases and Yn increases then is monotone. 38 Prove or give a counterexample: if product (xnYn) is monotone. 39 Prove: If Xn is a constant sequence if and only if increasing and monotone decreasing. 40 Let Xn = (-~)n. Find inf(Tn), sup(Tn), limsupxn, and liminf:rn.