1. Let x, a € R. Prove that if a <a, then -a < x <a.
let a,b > 0 . Prove that DI < Val
Let ne Nj. Prove that n < 2(6(n)).
Let U ? Rmxn. Prove that if UTI-In, then n < m.
(b) Suppose that en is a sequence such that 0 <In < 2011 for all n e N. Does lim an exist? If it exists, prove it. If not, give a counterexample. (c) Suppose that in is a sequence such that 0 < < 21 for all n E N.Does lim exist? If it exists, prove it. If not, give a counterexample. 20
Let n ez, n > 0; let do, d1,..., dn, Co,..., En be integers in the range {0, 1, 2, 3,4}. Prove: If 5*dx = 5* ex k=0 k=0 then ek = =dfor k = 0,1,...,n.
Let A, B, C be subsets of U. Prove that If C – B=0 then AN (BUC) < ((A-C)) UB
Prove AB _ BC__ AC aven. DEEF DE Prove: <A><D
Let X be a continuous random variable. Prove that: P(21-; < X < xạ) = 1 - a.
all three questions please. thank you Prove that for all n N, O <In < 1. Prove by induction that for all n EN, ER EQ. Prove that in} is convergent and find its limit l. The goal of this exercise is to prove that [0, 1] nQ is not closed. Let In} be a recursive sequence defined by In+1 = -) for n > 1, and x = 1. Prove that for all ne N, 0 <In < 1....