Real Analysis Answers

prove that the sequnce <an> defined by an=3n+7/4n+8 is a monotonic sequence.

prove that [1/2(a+b)]^2<=1/2(a^2+b^2)

Give an example of function that is differentiable, its derivative is positive at some point P but there is no neighboorhood of P such that function is monotone there.

Prove that if x is a rational number and y is an irrational number, then x + y is an irrational
number. If, in addition, x is not equal to 1, then show that xy is an irrational number.

if a,b,c are real numbers show that min{a,b,c}=min{min{a,b},c}

using principle of mathamatical induction prove that a+(a+d)+(a+2d)+...........+[a+(n-1)d] =
n/2[2a+(n-1)d]

Show that the closure of a set equals the union of the set and its complement.

Is their a connection between Dzeta function and prime numbers?
If there is, then show it, else explain why.

show that the union of finite set is finite

Find all real numbers x that satisfy the following inequalities.
1<x2<4