Real Analysis Answers

Prove that a strictly decreasing function is always one-one.

Show that 1/(n²+n+1) n belongs to N is a Cauchy sequence

nLet (a

Let (a_n) n belongs to N be any sequence. Show that lim_n--->∞ a_n= L iff for every ε > 0 there exists some N belongs to N

such that n ≥ N implies a_n belongs to N_ε (L)

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.

Give an example for each of the following.

i) A set in R with a unique limit point.

ii) A set in R whose all points except the one are its limit points.

iii) A set having no limit point.

iv) A set S with S^°= S̅

v) A bijection from N_odd to Z

If a sequence is bounded, then it has at least two convergent subsequences. Check whether the given statement is true of false. Give a counterexample in support of your answer.

Give an example for each of the following. (10)

i) A set in 

R

with a unique limit point.

ii) A set in 

R

whose all points except the one are its limit points.

iii) A set having no limit point.

iv) A set 

S

with 

S°=S.



v) A bijection from 

N odd

to 

Z

Which of the following statement s are true and which are false?Justify your answer 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.

ii)for every finite set S,sup S€S.

iii) There exists an interval with infimum and supremum equal.

iv)The sequence (1,1/2,1/3,1/4,....) is unbounded.

v) If a sequence is bounded ,then it has at least two convergent subsequences.

If a_(1)<**<a_(n) find the minimum value of f(x)=sum_(i=1)^(n)(x-a_(i))^(2) .