## 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