Equivalent conditions to a set being bounded/Statement

Statement

Let $(X,d)$ be a metric space and let $A\in\mathcal{P}(X)$ be an arbitrary subset of $X$ . Then the following are all logical equivalent to each other^{[Note 1]}:

$\exists C<\infty\ \forall a,b\in A[d(a,b)<C]$ - $A$ is bounded (the definition)
$\forall x\in X\exists C<\infty\forall a\in A[d(a,x)<C]$ ^[1]

Notes

Jump up ↑
Just in case the reader isn't sure what this means, if A and B are logically equivalent then:
- $A\iff B$ . In words " $A$ if and only if $B$ "

References

Jump up ↑ Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha

[1] Jump up ↑ Just in case the reader isn't sure what this means, if $A$ and $B$ are logically equivalent then:
$A\iff B$ . In words " $A$ if and only if $B$ "

[2] $A\iff B$ . In words " $A$ if and only if $B$ "

[FAVIDMH-2] Jump up ↑ Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha

[Note 1]

[1]

Equivalent conditions to a set being bounded/Statement

Statement

Notes

References

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