Real Analysis Answers

Find the limit of Xn=1/2[Xn-1+Xn]

The f:[ 3,13] to R defined by f(x)= [ x-3]- 13 is continuous function. True or false with full explanation

Show that the set

X= [1,5]U[6,9]U[15,30] is an compact subset of R

Every montonically increasing sequence are convergent. True or false with full explanation

Let fn(x) = cosnx/√ n , x belong to R, is not pointwise convergent. True or false with full explanation

Let f(x)= cos 1/x is not uniformly continuous on (0, infinity)

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]