Real Analysis Answers

The series

1/3+ 1/7+ 1/11 + 1/15+.....

is a convergent series.

True or false with full explanation

An integrable function can have finitely many points of discontinuous.

True or false with full explanation

The set Q of a rational numbers is a closed subset of R.

True or false

Test the following series for convergence,

Σ n x^n-1 , x > 0

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