Advanced Calculus 1: homework # 1Due day: Tuesday, September 20, 2011 recitations.All the problems are for 10 points. You have to solve all the problems and writesolutions in a legible way. Solutions that are hard to read will not be graded.Solution to each problem has to be written on a separate sheet of paper. The statementof the problem has to be included. Solutions that do not satisfy this conditionwill not be graded. Only randomly selected problems will be graded. I will announcewhat problems on the day after the due day. You need to show all your work. Answeris not enough.1. Let A be the set of all natural numbers that can be defined using no more than 100words. This is a finite set and hence there is the largest number in A. The number “thelargest number that can be defined using 100 words plus 1” is certainly defined withless than 100 words, but it is not in the set A, so we have a contradiction. Explain it.2. Prove that the set of all sets does not exist.3. Prove that if f : X → Y is a function and A1, A2, A3, . . . are subsets of X, thenf ∞[i=1Ai!=∞[i=1f(Ai) ,andf ∞\i=1Ai!⊂∞\i=1f(Ai). (1)Provide an example to show that we do not necessarily have equality in (1)4. Prove that if f : X → Y is one-to-one and A1, A2, A3, . . . are subsets of X, thenf ∞\i=1Ai!=∞\i=1f(Ai) .5. Prove that P (IN) has the same cardinality as IR.6. Prove that 1 +1√2+1√3+ ··· +1√n≥√n.7. Prove the binomial formula.8. Let a1, . . . , an, b1, . . . , bnbe positive numbers. Prove thatnYi=1(ai+ bi)1/n≥nYi=1a1/ni+nYi=1b1/ni.9. Find sup A and inf A, whereA =(n2+ 2n − 3n + 1: n = 1, 2, 3, . . .).10. Prove that if a > 1, then the functionQ 3 q 7→ aqis strictly increasing.11. Find the limit limn→∞ne1+12+···+1n.12. Find the limit limn→∞n sin(2πen!).13. Find the limitlimn→∞n3qn2+√n4+ 1 − n√2.14. Find the limit limn→∞sin2π√n2+ 1.15. Prove that the sequence sin n is