Check whether the intervals [2.5] and [7,10] are equivalent or not.

Show that the sequence (a), where   is monotonic. Is  Cauchy sequence  Justify your answer.

State Bonnet's mean value theorem for integrals. Apply it to show that:

Check, whether the collection given by:  open cover of 

Check whether the function  given by:

has local maxima and local minima.

Test the series:

 for absolute and conditional convergenc

Give an example of a series ∑ such that  ∑ is not convergent but the sequence converges to 0.

Show that on the curve, the chord joining the points whose abscissa are x=1 and x=2. is parallel to the tangent at the whose abscissa is

Using the sequential definition of the continuity, prove that the function defined

by: is discontinuous at each real number.

Let (a) be a sequence defined as show that (a_n) converges 

For the function, defined over [1,5], verify:  where P is the partition which divides [1.5] into four equal intervals. 

Show that

(i) 

(ii)

Give an example of an infinite set with finite number of limit points, with proper justification.

Show that the series   is uniformly convergent in  for any a>0.

Examine the function, for extreme values

Prove that continuous function of a continuous function is continuous.

Prove that the sequence  is convergent where (_n) is a bounded sequence.

Prove that

Prove that the set of integers is countable.

Examine the convergence of the following series

(i)

(ii)