1. First prove the following lemma: If a monotone (increasing or decreasing) sequence a_n has a convergent subsequence, then a_n converges.
2. Let r_n be an arbitrary sequence of all rational numbers (i.e., r is a function from N onto Q).
5. Write out the left-hand and the right-hand Riemann sum approximations of the area under the graph of y = 1/x from x = 1 to x = j, where the width of each subinterval in the Riemann sum is 1. Show one of the approximations is an under-estimate of the area and the other is an over-estimate. Then show the difference between the two approximations is at most 1, for all j. Use this to show that the given sequence is bounded above.
Sec 2.1: 11. Suppose toward contradiction the sequence a_n doesn't converge to alpha. Show for some epsilon > 0 there is a strictly increasing sequence n_k of natural numbers such that the distance between a_(n_k) and alpha is greater than epsilon for all k.