An example of a sequence $\{a_n\}$ such that $a_n > a_{n+1} > 0$ for all $n$, but $\{a_n\}$ does not have a strictly positive limit $L$, is the sequence:

$a_n = \frac{1}{n}$

Explanation:

• Monotonicity: This sequence is strictly decreasing since: $a_n = \frac{1}{n}, \quad a_{n+1} = \frac{1}{n+1}$ and $a_n > a_{n+1} \quad \text{for all} \quad n.$ Therefore, $a_n > a_{n+1} > 0$ for all $n$.

• Limit: The limit of the sequence is: $\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1}{n} = 0.$ However, 0 is not a strictly positive limit. The sequence approaches 0 but never reaches a strictly positive number.

Thus, $\{a_n\} = \left\{\frac{1}{n}\right\}$ satisfies the condition that it is strictly decreasing and positive for all $n$, but its limit is 0, which is not strictly positive.

Would you like more details or have any questions?

Here are 5 related questions:

1. Can you find a different sequence that decreases to 0 but not as fast as $\frac{1}{n}$?
2. What is the definition of a strictly positive limit?
3. Can you find a sequence that oscillates but still satisfies $a_n > a_{n+1} > 0$?
4. How does the limit of a sequence relate to convergence?
5. Can a sequence with a positive lower bound still have a limit of 0?

Tip: When analyzing limits, always check whether the sequence approaches a positive or negative number, or if it diverges to infinity.