7. Show that An → A as n→∞ I{An} - → I{A} as n→ ∞.

15x4- 13x2+ a is o(x*). Hint: Apply theurem I. x"+1 is o(x). Hint: Use lng divisinm. 2. i"+a+3"+ --- +n" =(n"). Hint: For n>! 4 4 3. +n = O(n'). Hìnt: For n>I *sn)くn"etc 4. 2*+17 is O(3") . * , Hint: 2く3 and 17<3*, X23 5. Carefully explain what is by meant 6. Find n where (x²+ ax+1)* is o(X). Hint: expand (x*+ax+U and apply theorem 1.

[о, 1] x [1, 2]

Prove this proposition algebraically

Let C(x, y) mean that student x is enrolled in class y, where the domain for x consists of all students in your school and the domain for y consists of all classes being given at your school. Express EXayvz((x = y) ^ (C(x, z) → C(y, z))) by a simple English sentence. A) There exist two distinct people, the second of whom is enrolled in every course that the first is enrolled in. В There exist two distinct people enrolled in exactly the same courses. c) Some student is enrolled simultaneously in two courses. D Someone is enrolled in all courses.

Explain the Proof of each of the following in details

U(5) is not isomorphic to U(10) اختر احدى الدجابات True False

7. Let F(x) = f(x) = x" for xe [a, b] and some positive odd integer n. Show * JdF 80². 2 a =