Question

Suppose you know that\(\left\{ {{{\bf{a}}_{\bf{n}}}} \right\}\)is a decreasing sequence and all its terms lie between the numbers 5 and 8 . Explain why the sequence has a limit. What can you say about the value of the limit?

Short answer

\(5 \le L < 8\)

Step-by-step solution

Monotonic sequence theorem

Every bounded, monotonic sequence is convergent.

Observation

Given that \(\left\{ {{a_n}} \right\}\)is decreasing sequence

So \({a_n} > {a_{n + 1}} > {a_{n + 2}} > {a_{n + 3}} > \ldots \ldots .\)for all\({\rm{n}} \ge 1\).

\(\left\{ {{a_n}} \right\}\)is bounded sequence, since all terms lie between 5 and 8 .

Then by Monotonic sequence theorem \(\left\{ {{a_n}} \right\}\)is convergent.

So \(\left\{ {{a_n}} \right\}\) has a limit say\(L\).

Since 8 is an upper hound of\(\left\{ {{a_n}} \right\}\), so\(L\)must be less than 8

Since \(\left\{ {{a_n}} \right\}\)is decreasing sequence so \(5 \le L < 8\).