Q. 15 - Calculus

Question

Give examples of sequences satisfying the given conditions or explain why such an example cannot exist.

A decreasing sequence that is bounded below but is not bounded above.

Step-by-Step Solution

Answer

Examples of the sequences is $\left\{a_k\right\}$.

Step 1. Given information.

Consider the given question,

A decreasing sequence that is bounded below but is not bounded above.

Step 2. Take the sequence a k .

Consider the sequence $\left\{a_k\right\}$.

The sequence is decreasing if $a_{k+1}\le a_k$ for $k>0$.

The decreasing sequence is always bounded above because first term of the sequence is the upper bound of the sequence is decreasing sequence.

So, there is no decreasing sequence which not bounded above.

Hence, this sequence is decreasing sequence that is bounded below but is not bounded above.