Determine if the given set is Bounded- Complex Numbers

[chwala]

Homework Statement

See attached.

Relevant Equations

Complex Numbers

My interest is only on part (a). Wah! been going round circles to try understand why the radius = $2$. I know that the given sequence is both bounded and monotonic. I can state that its bounded above by $1$ and bounded below by $0$. Now when it comes to the radius=$2$, i can also say that the boundary of set $S$ also consists/ includes the complement of the set and that will gives us;$r^2=\sqrt{(1--1)^2+(0-0)^2}=\sqrt{4}$
$r=2.$

I hope this is the correct reasoning, otherwise i need your insight...i also tried looking at the cauchy criterion,... among other...

 

[anuttarasammyak]

Say s is a member of S other than i , as $|s| < 1$ s is inside unit circle of complex plane on which i is. Both s and i are inside the circle |z|=2.

Not only |z|$\leq$2 but other regions including S inside work, e.g. |z|$\leq$3/2, |z|$\leq$3,
the rectangle region of $0 \leq I am \ z \leq 1+\epsilon_1, |Re\ z| \leq \epsilon_2$ where $0<\epsilon_1,\epsilon_2$.

 

[chwala]

Say s is a member of S other than i , as $|s| < 1$ s is inside unit circle of complex plane on which i is. Both s and i are inside the circle |z|=2.

Not only |z|$\leq$2 but other regions including S inside work, e.g. |z|$\leq$3/2, |z|$\leq$3,
the rectangle region of $0 \leq I am \ z \leq 1+\epsilon_1, |Re\ z| \leq \epsilon_2$ where $0<\epsilon_1,\epsilon_2$.
View attachment 316045

Something am not getting here, I guess I need to check...I thought the set members are bound by $1$ and $0$. You have $1.5$ which clearly in my understanding is not part of set S. My understanding of radius here is the distance between i.e neighbourhood of set $S$ and $S^{'}$.

 

[anuttarasammyak]

You have 1.5 which clearly in my understanding is not part of set S.

You are right. Even as for the answer in the textbook, z=1 belongs to |z|<2, but it does not belong to set S.

Say T is a set, e.g. |z| $\leq$ 2, |z| $\leq \sqrt{2}$, |z| $\leq \pi$, |z| $\leq 3/2$ ... any circle larger than unit circle in complex plane,
$S \subset T$
Set T is not part of set S. Set S is part of set T.