**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

.

**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.
*
- If
is an increasing sequence and is bounded above, then it is convergent.
- 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