Need help with algorithm analysis

Please help me to understand the attached questions.

Proof by Contradiction

Prove the following fact (give a proof by contradiction): There do not exist constants N > 0 and C > 0 such that ∀n ≥ N ,n^2 ≤ C*n

Prove that the big-O relationship is transitive

Prove that the big-O relationship is transitive (give a direct proof). That is, if f(n), g(n) and h(n) are positive-valued functions: if f(n) is Ο(g(n)) and g(n) is Ο(h(n)) then f (n) is Ο(h(n))

Child/Sibling implementation

Child/sibling implementation for following tree/graph!

Divide and Conquer Binary Tree

Consider an n-node complete binary tree T, where n=2^d - 1 for some d. Each node v of T is labeled with a real number x_v. You may assume that the real numbers labeling the nodes are all distinct. A node v of T is a local minimum if the label x_v is less than the label x_w for all nodes w that are joined to v by an edge. You are given such a complete binary tree T, but the labeling is only specified in the following implicit way: for each node v, you can determine the value x_v by probing the node v. Show how to find a local minimum of T using only O(log n) probes to the nodes of T. (This is a problem from the book Algorithm Design by Kleinberg and Tardos, page 248, problem 6)