View source for Bounded set

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==Definition==
Let {{M|(X,d)}} be a [[metric space]]. Let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset of]] {{M|X}}. Then we say "{{M|A}} is bounded in {{M|(X,d)}}" if{{rFAVIDMH}}:
* {{M|1=\exists C<\infty\ \forall a,b\in A[d(a,b)<C]}} - where {{M|C}} is real<ref group="Note">{{M|C\in\mathbb{R}_{\ge 0} }} should do as {{M|0}} could be a bound, I suppose on a one point set?</ref>
{{anchor|unbounded}}If a set is not bounded it is "''[[unbounded]]''" (that link redirects to this line)
==[[Equivalent conditions to a set being bounded|Equivalent conditions]]==
{{:Equivalent conditions to a set being bounded/Statement}}
==See also==
* [[Diameter (set)]]
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Metric Space|Functional Analysis|Analysis}}
=OLD PAGE=
==Notes==
{{Todo|Surely this can be generalised to an arbitrary [[Metric space]]}}

==Of {{M|\mathbb{R}^n}}==
Given a set {{M|A\subseteq\mathbb{R}^n}} we say {{M|A}} is bounded<ref name="Mendelson">Introduction to topology - Bert Mendelson - Third Edition</ref> if:
* {{M|\exists K\in\mathbb{R} }} such that {{M|\forall x\in A}} (where {{M|1=x=(x_1,\cdots,x_n)}}) we have {{M|\vert x_i\vert\le K}} for {{M|i\in\{1,\cdots,n\} }}

==Immediate results==
{{Begin Theorem}}
(Real line) {{M|A\subseteq[-K,K]\subset\mathbb{R} }} (where {{M|K> 0}} and {{M|[-K,K]}} denotes a closed interval) ''if and only if'' {{M|A}} is bounded. 
{{Begin Proof}}
This follows right from the definitions, the {{M|K}} is the bound. 

'''Proof:''' {{M|\implies}}
: If such a {{M|K}} exists then it is the bound. We are done.

'''Proof:''' {{M|\impliedby}}
: If it is bounded then it is easy to see that given a bound {{M|K}}, {{M|A\subseteq[-K,K]}} (use the [[Implies and subset relation|implies-subset relation]])


''Note:''<ref name="Mendelson"/> stipuates the {{M|A\subseteq[-K,K]\implies}} bounded direction only.
{{End Proof}}{{End Theorem}}
{{Begin Theorem}}
(Real line) Every closed interval ({{M|[a,b]}} for {{M|a,b\in\mathbb{R} }} and {{M|a\le b}}) is bounded.<ref name="Mendelson"/>
{{Begin Proof}}
Choose {{M|1=K=\text{Max}(\vert a\vert,\vert b\vert)}} - result follows
{{End Proof}}{{End Theorem}}

==See also==
* [[Topological property theorems]]

==References==
<references/>

{{Definition|Topology|Metric Space}}