Real Analysis Answers

Are the statement true or false?give reason for your answers: the function,f(x) =sin^2x is uniformly continuous in the interval [0,π]

Are the statement true or false?give reason for your answers: the function f defined on r by f(x) ={0 x is rational 2 is irrational} Is integrable in the interval [2,3]

Let a and b be two cardinal numbers. Modify Cantor’s definition of a < b to define a ≤ b. (Hint: Examine what happens if you drop condition (a) from Cantor’s definition of a < b.) 2. Prove that a ≤ a. 3. Prove that if a ≤ b and b ≤ c, then a ≤ c. 4. Do you think that a ≤ b and b ≤ a imply

a = b? Explain your reasoning. (Hint: This is not as trivial as it might look.)

verify cauchy's mean value theorem for the function f(x) = x and g(x) = sinx in [0,π/2].

Check the convergence of the sequence defined by 𝑢𝑛+1 = 𝑎 /(1+𝑢𝑛) where 𝑎 > 0, 𝑢1 > 0.

F(x)=1-2x

1. Find the fourier cosine Series for the function f(x)=x^2 for 0<x<π
2. Obtain a fourier expression for f(x)=x^2 for -π<x<π where f(x) is a even function

Find the limit superior and the limit inferior of the following sequences

a) {(1 +

1

𝑛

)

𝑛+1

}

b) {

(−1)

𝑛

𝑛2

}