The problem here is to prove that the recurrence relation $T(n) = T\left(\left\lfloor \frac{n}{2} \right\rfloor \right) + 1$ is $O(\log n)$. There are two attempts shown in the image to prove this claim.

Attempt #1:

1. The goal is to show that $T(n) \leq c \log n$.
2. The user tries to apply the recurrence and uses a step like $T(n) \leq c \log \left\lfloor \frac{n}{2} \right\rfloor + 1$.
3. Then, they approximate $\left\lfloor \frac{n}{2} \right\rfloor \leq \frac{n+1}{2}$, leading to: $T(n) \leq c \log \left( \frac{n+1}{2} \right) + 1 = c \log (n+1) - c + 1.$
4. This step leads to $T(n) \leq c \log (n+1)$, which seems incorrect because it doesn't yield the desired $c \log n$, as indicated by the sad face.

Attempt #2:

1. This attempt tries a more refined analysis with an offset $\alpha$, where $T(n) \leq c \log (n - \alpha)$ for some $\alpha \in \mathbb{R}$.
2. The recurrence is rewritten as: $T(n) \leq c \log \left\lfloor \frac{n}{2} \right\rfloor - \alpha + 1.$
3. Again, approximating $\left\lfloor \frac{n}{2} \right\rfloor \leq \frac{n+1}{2}$, leads to: $T(n) \leq c \log \left( \frac{n+1}{2} - \alpha \right) + 1.$
4. After manipulating terms and assuming $\alpha \geq 1$, the result is: $T(n) \leq c \log (n - \alpha).$
5. The user emphasizes checking the base case, which is necessary to verify the correctness of the assumption $T(n) \leq c \log n$ at small values of $n$.

Conclusion:

The second attempt seems more promising due to the introduction of an offset $\alpha$, which handles the behavior of the recurrence more carefully. Ensuring the base case works is crucial to finalizing the proof.

Would you like further clarification on any part of this explanation, or a more detailed breakdown of the steps?

Here are 5 questions you might consider:

1. What is the role of the constant $c$ in the analysis?
2. How does the floor function $\left\lfloor \frac{n}{2} \right\rfloor$ affect the recurrence?
3. Why does the first attempt fail to give the desired bound?
4. How can checking the base case help ensure the overall proof is correct?
5. Could other methods (e.g., recursion trees) help in showing the recurrence is $O(\log n)$?

Tip: When proving recurrences, carefully tracking constants and small changes to $n$ can often make the difference between a correct and incorrect bound.