View source for Equivalent conditions to a set being bounded

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==[[/Statement|Statement]]==
{{/Statement}}
==Proof of claims==
{{begin Inline Theorem}}
[[/1 implies 2|{{M|1\implies 2)}}]] {{M|1=\big(\exists C<\infty\ \forall a,b\in A[d(a,b)<C]\big)\implies\big(\forall x\in X\exists C<\infty\forall a\in A[d(a,x)<C]\big)}}, that boundedness implies condition 2
{{Begin Inline Proof}}
{{/1 implies 2}}
{{End Proof}}
{{End Theorem}}
{{begin Inline Theorem}}
[[/2 implies 1|{{M|2\implies 1)}}]] {{M|1=\big(\forall x\in X\exists C<\infty\forall a\in A[d(a,x)<C]\big)\implies \big(\exists C<\infty\ \forall a,b\in A[d(a,b)<C]\big)}}, that condition 2 implies boundedness
{{Begin Inline Proof}}
{{/2 implies 1}}
{{End Proof}}
{{End Theorem}}
==Notes==
<references group="Note"/>
==References==
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{{Theorem Of|Analysis|Functional Analysis|Topology|Metric Space}}