1. ## Boundedness in R^k

Hello everyone!

I'm new to the amazing world of analysis. I've got a question related to boundedness.
A set $A$ is bounded in $R$ if there exists two real numbers $a$ and $b$ such that $A\subset ]a;b[$.

Well since we're talking about $R^1$, we can't differentiate between and open ball and a 1-cell, but when we're talking about boundedness in the complex plane $R^2$ for instance, must the set $A$ of points $P(x,y)$ be $a_1 < x < a_2$ and $b_1 < y < b_2$ or is it that $d(p,q) = |p-q| < r$ for some $q$??

Thanks!!

2. Originally Posted by rebghb
Hello everyone!

I'm new to the amazing world of analysis. I've got a question related to boundedness.
A set $A$ is bounded in $R$ if there exists two real numbers $a$ and $b$ such that $A\subset [a,b][$.

Well since we're talking about $R^1$, we can't differentiate between and open ball and a 1-cell, but when we're talking about boundedness in the complex plane $R^2$ for instance, must the set $A$ of points $P(x,y)$ be $a_1 < x < a_2$ and $b_1 < y < b_2$ or is it that $d(p,q) = |p-q| < r$ for some $q$??

Thanks!!
There are multiple definitions. I mean, all you need is that $A$ is contained in some bounded set. The real definition of boundedness is $A\subseteq M\text{ is bounded}\Leftrightarrow \text{diam }A<\infty$ for $\mathbb{R}^m$ (and other normed vector spaces) it is easier to think that $A\text{ is bounded }\Leftrightarrow \|a\| for some fixed $K<\infty$. So, $A$ is bounded if it's either contained in some open ball $B_{\delta}(0)$ or it's contained in some $n$-cell $[a,b]^n$