Question

Prove that every convergent sequence is a cauchy

Answer 1

Answer:

The proof is essentially the same as the corresponding result for convergent sequences. Any convergent sequence is a Cauchy sequence. If (an)→ α then given ε > 0 choose N so that if n > N we have |an- α| < ε. Then if m, n > N we have |am- an| = |(am- α) - (am- α)| ≤ |am- α| + |am- α| < 2ε