Real Analysis Answers

Give an example for each of the following:

A set in R with a unique limit point.

Which of the following statements are true and which are false? Justify your answers with

a short proof or a counter-example:

For every finite set S, S ∈ S

Which of the following statements are true and which are false? Justify your answers with

a short proof or a counter-example:

If x and y are real numbers such that x < y, then x²<y²

Show that the function f defined on [0,1] by f(x)= (-1)^(n-1) for 1/(n+1) < x/n ≤ 1/n where (n=1,2,3...) is integrable on [0,1]

Test whether the series ∞Σn=0 1/(n^5+x^3) converges uniformly or not

If F is Lipschitz function and g(x) is monotonically increasing on [a,b] then fog is bounded variation

Consider the function

F(x)= xsin(π/x) 0<x≤1

0 x=0

Show that f(x) is continuous but not of bounded variation

What is the limits of (1/n)

Test whether the series ∞Σn=0 1/(n^5+x^3) converges uniformly or not

Show that the function f defined on [0,1] by f(x)= (-1)^(n-1) for 1/(n+1) < x/n ≤ 1/n where (n=1,2,3...) is integrable on [0,1]