Cauchy Sequence

A sequence is Cauchy iff:

Bounded

All Cauchy sequences are bounded. (has an upper bound).

Proof Hint

• Consider the Cauchy definition
• Take

Convergence & Cauchy

A sequence is converging iff it is Cauchy.

Proof Hint

To prove :

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

To prove :

• Consider the definition of Cauchy sequences
• Show that the sequence is bounded

Complete

A set is complete iff:

IMPORTANT: is not complete because:

IMPORTANT: is complete.

Proof Hint

Proof is quite hard.