1. For this problem, review the material in Appendix A on logic and logical notation. For each of the following statements, (i) convert the statement to logical notation; (ii) negate the statement in logical form; and (iii) convert the negation back to a mixture of words and symbols similar to the original statement. (a) For every real number a, there is an 15 greater than zero such that there is a real number M such that for all a: E (ae, G-I-E), |f(33)| g M. (b) [It is understood that A and B are sets.] There exists a function f : A > B such that for every $.31 E A, :1; 7 y, there is at least one z E A such that f(z) 7 x) and f(z) 7E y). (c) For every 6 > 0, there is a 6 > 0 such that, for all 3.3; 6 IR, |f(33) - f(y)| 5 Elm yl's- 2. Let S C R be an innite set which is bounded below. Let 1:} = inf S. Prove that there exists a decreasing sequence X 2 (can) in S such that limX = w. 3. Suppose that Y = (yn) is a convergent sequence of real numbers. Prove that for any subsequences of Y, Y' := (yak) and Y\" := (ymk), lynk ymk| converges to 0