Real Analysis Answers

Suppose f is continuously differentiable on [0,1] and f'' (greater than or equal to) 0 on [0,1]. Prove that f(x) (greater than or equal to) f(c) + f'(c)(x-c) for every x,c in [0,1].

If f from R to R is continuous and open, show that f is strictly monotone.

If ∑ an with an > 0 is convergent, then is ∑ (an an+1)1/2 always convergent? Either prove it
or give a counterexample.

1. Consider the sequence
x1 = 1, x2 = 1 - 1/2, x3 = 1 - 1/2 + 1/4, x4 = 1 - 1/2 + 1/4 - 1/8...

(a) Show that:

(i) The subsequence x1; x3; x5; : : : is decreasing.

(ii) The subsequence x2; x4; x6; : : : is increasing.

(iii) Each even-numbered term is less than each odd-numbered term.

Show that the convergence of a series is not affected by changing a finite number of its terms

The function f : R to R+ U {0} , f(x) : x^2 is one-one
but not onto.

State true or false, why?

Give an example of sequence (Yn) and (Zn) for which limit as n aproaches to infinity Yn/Zn equals one, but (*) does not hold

Prove that if the sequence (Un) is convergent and bounded above by M, then the limit is bounded above by M

Use a contradiction argument to prove that 3√2 is irrational