Real Analysis Answers

the period of the function f(x)= |cos^3 3x| is pi/3. True or false with full explanation

Let fn(x)= n/(x+2n) is uniformly convergent in [0,k] , k>0

Check whether the function f given by

f(x)=1/(2x-4)^2 ∀x∈]-2,2[

Is continuous in the interaval ]-2,2] but not bounded in the interval (-2,2)

Use Weierstrass’ M-Test to prove that the series

Infinity

∑ n^2x^n

n=1

converges uniformly in the interval [0,1/5]

Prove that if f and g are Riemann integrable on [a, b], then f · g and f + g are Riemann integrable on [a, b].

Prove that a subset of a set of measure zero has measure zero.

Show that the union of two sets, each of measure zero, has measure zero

Let fn(x)= nx/(1+nx) is not uniformly convergent on [0,1]

Let fn(x)= x^n is not uniformly continuous on [0,1] but is uniformly continuous on [0,k]

Check the whether the set { 2/7, 2/8, 2/9,..} is countable or not. Also a give example of proper subset of R which is uncountable