ANALYSIS I
Properties of Limits, Cauchy sequences
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Theorem. Suppose that limn®¥ an = a , then prove that limn® ¥|an| = |a| .

Theorem. Convergent sequences are bounded.

Defn.  A sequence {an} is called monotone increasing if am £ an whenever m £ n. A sequence {an} is called monotone decreasing if an£ am whenever m £ n.

Theorem. Monotone sequences, which are also bounded, converge.

Theorem. Suppose that limn®¥an = a and limn®¥bn = a. If an £ cn£ bn for all n Î IN, then limn®¥cn exists and equals a.

Theorem.(Properties of Limits) Suppose that limn®¥an = a and limn®¥bn = b, then

  1. limn®¥an + bn = a+b
  2. limn®¥an bn = a b
  3. If b ¹ 0, then limn®¥ [(an)/(bn)] = [a/b].
Defn.  A sequence {an} is called Cauchy if for each e > 0 there is an N Î IN so that |am - an|< e whenever N £ m,n.

Theorem. Each convergent sequence is Cauchy.

Theorem. Each Cauchy sequence is convergent.

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