We will prove the Cauchy-Schwarz inequality for infinite series. Let (a,) and (bm) be two sequences such that a and E both converge. (a) Give an example a sequence (a„) where Ea, diverges but E¢ converges. (b) Use induction to show that for any N > 0 N This is the Cauchy-Schwarz inequality for finite sums (or the dot products). (c) Show the (Ena)" (E«) (E•) Σ This is the Cauchy-Schwarz inequality for infinite sums. (d) Why does part (c) imply that ab, converges?

If 0 < r < 1 and {n} is a sequence with the property |xnxn+1| < pn for n = 1,2,3,, then does {n}=1 converge or diverge? Prove your assertions.

a) Suppose (an) is Cauchy and that for every k∈N, the interval (−1/k,1/k) contains atleast one term of (an). Can we say that (an) converges to 0? Either show that it does or give a counter-example. b) Suppose (xn) is Cauchy and (yn) is bounded and monotone. Can we say anything aboutwhether the sequence (xn yn) converges or not? Justify your answer!

Determine whether the sequence converges or diverges n! An (п + 2)! [ Select ] In(2 + e") 3n