Proof That the Diameter of a Set Equals the Diameter of its Closure
Theorem
Ifis a subset ofin a metric spacethenwhereis the diameter of
Proof
Take
Supposethen
Choosesuch that
From the triangle inequality,
Henceso
Since
Hence
Theorem
Ifis a subset ofin a metric spacethenwhereis the diameter of
Proof
Take
Supposethen
Choosesuch that
From the triangle inequality,
Henceso
Since
Hence