Real Analysis Answers

The function: f : [-1,3]→R defined by: f(x)= 3x+1/(x^2+4) is uniformly continuous on [-1,3].

True or false with full explanation

Give an example of a divergent sequence which has two convergent sequences. Justify

your claim.

Which of the following statements are true and which are false? Justify your answers with 

a short proof or a counter-example.

i) If 

x

and 

y

are real numbers such that 

x < y,

then

x^2 < y^2.

The product of two divergent sequences is divergent. True or false? Justify.

The product of two divergent sequences is divergent. True or false? Justify.

Give an example of a divergent sequence which has two convergent sequences. Justify  your claim

lim n tend to infinity [1/√2n-1 + 1/√4n-2² +1/√6n-3²+ .......+ 1/n] = π/2

Check whether the intervals [2,5] and [7,10] are equivalent or not

Show that the sequence (an), where an= n/n^2+4 is monotonic. Is (an) a Cauchy sequence? Justify your answer

The function: f:[-1,3] →R defined by:

f(x)= 3x+1/x^2+4 is uniformly continuous on [-1,3].

True or false with full explanation