**Given:**

The sequence
{an}
is a decreasing sequence.

All terms in the sequence
{an}
lies between 5 and 8.

That is,
5≤an<8
.

**Definition used:**

(1) If the sequence is either increasing or decreasing, then the sequence is called monotonic.

(2) If
{an}
is the sequence with
m≤an≤M
for
m,M∈ℕ
, then the sequence is bounded.

**Theorem used: Monotonic Sequence Theorem**

“If the sequence is bounded and monotonic, then the sequence is convergent.”

**Calculation:**

Since the sequence
{an}
is a decreasing sequence,
an>an+1
for all
n≥1
.

Since
5≤an<8
and by definitions (1) and (2),
{an}
is monotonic and bounded.

Thus, it can be concluded that the sequence has a limit.

Obtain the value of the limit.

Use the Monotonic sequence Theorem, the sequence
{an}
is convergent.

That is,
limn→∞an=L
. (1)

Consider
5≤an<8
.

Apply limit on both the sides.

limn→∞(5)≤limn→∞(an)<limn→∞(8)5≤limn→∞(an)<85≤L<8 [By equation (1)]

Thus, the limit of the sequence can by any number between 5 and 8.

Therefore, the value of the limit is
5≤L<8
.