Real Analysis Answers

Find the following limit.:

Lim x →0 for 1-cos x²/x².sin x²

Prove the limit of 𝑥𝑛 =1/2 [𝑥𝑛−1 + 𝑥𝑛]

let ck>0 for k=1,2,....,n.show that( c1+c2+....cn)/√n≤[ c1^2+c2^2+....+cn^2]^1/2≤c1+c2+...+cn

Find all real numbers x such that

1<x^2<4

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)