provide an example of a sequence {an} such that an > an+1 > 0 for all of n but {an} does not have a strictly positive limit L. 

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.

Provide an example of a sequence {an} such that an > an+1 > 0 for all n but {an} does not have a strictly positive limit L.

Explore a sequence {an = 1/n} that is strictly decreasing and positive for all n, yet converges to 0, which is not a strictly positive limit. Learn the concepts of monotonicity and limits, perfect for undergraduate calculus or real analysis students.

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