# Ordering and convergence.

Proposition 3.8.1   Let be a sequence whose terms are all non negative. If this sequence is convergent, with limit , then .

Example 3.8.2   Let . Then . We have .

Corollary 3.8.3   Let and be two sequences such that for any natural , (or ). If both sequences are convergent, with respective limits and , then .

Example 3.8.4   Let and . Then . We have .

Proposition 3.8.5 (Sandwich theorem)   Let , and be three sequences whose terms verify the condition: . If this sequences and are convergent, with the same limit , then is convergent and its limit is equal to .

Example 3.8.6   Let . Then for any natural , we have: . As , we have .

Corollary 3.8.7   If the sequence converges towards the limit , then the sequence converges towards the limit .

The converse of Cor. 8.7 is generally false: for example, if , then is constant, thus convergent, but is periodic with period 2, thus divergent.

For , the converse of Cor. 8.7 is true.

Proposition 3.8.8   Let be a sequence of real numbers.
1. If is an increasing sequence and is bounded above, then it is convergent.
2. If is a dereasing sequence and is bounded below, then it is convergent.

Example 3.8.9   Let be defined by .
• By induction, prove that .
• By induction, prove that is an increasing sequence.
Thus, is convergent.

Noah Dana-Picard 2007-12-28