Real Analysis Answers

Let A be non empty sebset of real number which is bounded above let -A be the set of all real number -x where x belong A show that sup(A)= -inf(-A)

If x and y belong to real numbers, show that
|x+y|/1+|x+y|<=|x|/1+|x|+|y|/1+|y|

If x and y are irrational number show that x+y and x*y are irrational.

Between any two distinct real numbers there exist an infinite many irrational numbers. Prove this statement.

Test the convergence of the series : 2/1^3 - 2/2^3 + 3/3^3 + 5/4^3 ........

prove by method of contradiction that α is an irrational number and ß is an rational number ,then α+ß is an irrational number.

If f(x) =root under x and phi(x) =1/root x in (a, b), then verify cauchy mean value theorem.

If f and g are differentiable on R and 1<=f'(x)<=2 for all x in R.
Prove that for every a in R we have
f(x)-f(a)<=g(x)-g(a) for all x in [a,infty).

Q1:
f is differentiable at c, prove that:
1) f′(c)=limh→0((f(c+h)−f(c)0\h)
2)f′(c)=limh→0(9f(c+h)−f(c−h))\2h)


Q2:
Let f:R→R .The function f is even if f(−x)=f(x) for all x∈R, and odd if f(−x)=−f(x) for all x∈R. if f is differentiable, prove that f′ is odd when f is even, and when f is odd.

If A is lebesgue measurable subset of R of positive measure and 0< δ< X(A) .then show that there exists a measurable subset B of A satisfying λ(B)=δ