1) If (bn) is a bounded sequence and lim(an) = 0, show that the lim(anbn) = 0. Explain why Theorem 3.2.3 cannot be used.
Note: Here's Theorem 3.2.3
(a) Let X = (xn) and Y = (yn) be sequences of real numbers that converge to x and y respectively, and let c be an element R. Then the sequences X+Y, X-Y, X∙Y, and cX converge to x+y, x-y, xy, and cx, respectively.
(b) If X = (xn) converges to x and Z = (zn) is a sequence of nonzero real numbers that converges to z and if z ≠ 0, then the quotient sequence X/Z converges to x/z.
2) Let x1 > 1 and xn+1 : = 2 - 1/xn (for n an element of N). Show that (xn) is bounded and monotone. Find the limit.
There are several examples of working with sequences and limits to complete broofs.