Recent content by v.rad

1. Suppose that f: (a,b) --> R is convex. Prove Jensen's inequality: if x1,...,xn\in(a,b) and c1,...,cn >= 0 s.t. \sum(c_j)f(x_j) >= f(\sum((c_j)(x_j)) both summations from j = 1 to n 2: Convex: whenever x1, x2 \in(a,b) and 0 <= c <= 1, we have cf(x1) + (1 + c)f(x2) >= f(cx1 + (1-c)x2)...

If a_n >= 0 for all n, and the series a_n converges, then n(a_n - a_n-1) --> 0 as n --> infinity. Prove or disprove the statement using a counterexample. I know that the statement is false...I am just having terrible difficultly finding a counterexample...

[b]1. sup (empty set) = -infinity, and if V is not bounded above, then sup V = +infinity. Prove if V\subseteqW\subseteqReal Numbers then sup V is lessthan/equalto supW [b]3. I used a proof by contrapositive, but I'm not sure if it is completely valid...