Proof That a Nested Local Base at a Point of a Topological Space Contains a Sequence Converging to the Point

Theorem

Letbe a topological space and letbe a nested local base at a pointLetbe a sequence such thatThenconverges to

Proof

Letbe an open set containing

is a local base athenceexists such that

Setsare nested if

Hence for anywithand sincewe have

Hence