Real Analysis Answers

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.

show if a,b ЄR then,
a. max (a,b) = (1/2)(a+b+|a-b|) and min (a,b)=(1/2)(a+b-|a-b|)
b. min (a,b,c)=min(min(a,b),c)

Find all x ∈ R that satisfy the equation& |x+1| + |x-2| = 7.

If& x, y, z& ∈ R and& x ≤ z, show that& x ≤ y ≤ z& if & |x-y| + |y-z| = |x-z|.& & Interpret this geometrically.