Real Analysis Answers

The first five terms of a sequence (x sub n) are given below.&
& 1/2,-1/4,1/8,-1/16,.....

assuming that the "natural pattern" indicated by these terms persist, give a formula for the nth term (x sub n).&

The sequence (x[sub]n[/sub]) is defined by the following formulas for the n[sup]th[/sup] term. Write the first five terms in each case.&
& (x[sub]n[/sub])= ((-1)[sup]n[/sup])/n

If [i]a,b[/i]& are element of real numbers, show that [b]|a+b|=|a|+|b|[/b] & if [b]ab>=0[/b]

Explain the method of finding sqare root of 5.

let A and B be bounded nonempty subset of R, and let A+B:={a+b:a element A,b element B}.
prove that the Sup(A+B)=Sup A + Sup B and inf (A+B)= inf A + inf B.

let S subset R be nonempty. show that if u= supremum S, then for every number n element N the number (u-1)/n is not an upper bound of S, but the number (u+1)/n is an upper bound of S.

If f(x)<=g(y) for all x,y element D, then we may conclude that the upremum f(D)<=infimum g(D), which we may write as: Sup f(x)<=inf g(y) such that x,y element of D

What is the interpretation of [x]?


What is the domain of function f(x)=1/[x]+1?

If a < x < b and a < y < b,& show that |x-y|< b-a. Interpret this geometrically.

Show that if a,b Є R, and a≠b, then there exist ε-neighborhoods U of a and V of b of such that
(U intersection V)= Ø.