Answer to Question #174833 in Real Analysis for Aarohi

Question

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

Answer 1

(a) Begin with any co nvergent sequence, and ,,spoil'' it in sufficiently many places to ensure it does not converge, but take care not to create additional limit points that way. You could for instance take the convergent sequence $c_n:=\dfrac{1}{n}$ and define:

 

$a_n = n$  if n is a power of 2 

   $=\dfrac{1}{n}$ if n is not a power of 2

(b) $S=(−1)^n+\dfrac{1 }{n} ∣n∈N$

(c) Set of natural numbers

(d) set of whole numbers

(e)  Let f:N→Z where

$f(n)=\dfrac{n}{2}, \text{ if n is even} \\$

$=\dfrac{−(n+1)}{2} \text{ ,if n is odd}$