Cauchy Sequence

A sequence is Cauchy iff:

Bounded

Converse is not true.

Proof Hint

• Consider the Cauchy definition
• Take

Convergence & Cauchy

Converse is true only when the sequence is a subset of a Complete set.

Proof Hint

• Consider the limit definition of converging sequences
• Introduce the converging value (say ) into the inequality and split into 2 parts

Complete

A set is complete iff:

IMPORTANT: is complete. Proof is quite hard.

IMPORTANT: is not complete because:

IMPORTANT: is complete.