Update app.py

app.py

@@ -584,9 +584,7 @@ STRICT REQUIREMENTS:
 7. At the end of the solution output, print SymPy code that you would use to solve or verify the main equations in the question 
 8. Observe the folloiwng SymPy Guidelines
 {SYMPY_GUIDELINES}
-9. For problems where the subject is Real Analysis and the question type is proof, observe the following guidelines:
-   ### Real Analysis Proof Guidelines  
-For Real Analysis proofs, follow these principles to ensure clarity, rigor, and logical completeness:  
+9. For problems where the subject is Real Analysis, observe the following guidelines:
 
 a. **Justify Every Step**  
    - Provide detailed reasoning for each step and explicitly justify every bounding argument, inequality, or limit claim.  
@@ -596,11 +594,12 @@ a. **Justify Every Step**
 b. **Handling Limits and Differentiability**  
    - In epsilon-delta proofs, clearly explain why the chosen delta works.  
    - When using limit substitutions, justify why the transformation preserves limit behavior.  
-   - If verifying differentiability, explicitly state why continuity at that point is necessary and how it connects to the derivative’s existence.  
+   - If verifying differentiability, explicitly state why continuity at that point is necessary and confirm that continuity has been established before proceeding.  
    - If proving continuity or differentiability, check symmetry in approach from both sides (left-hand and right-hand limits or derivatives).  
 
 c. **Function Properties and Integrability**  
    - If stating that a function is Riemann integrable, compact, or uniformly continuous, explain why.  
+   - If claiming a function is continuous for all x ≠ 0, explicitly justify why using function composition, bounded functions, or known theorems.  
    - When assuming an integral is finite, provide justification based on function class properties (e.g., Riemann integrability implies boundedness).  
 
 d. **Inequalities and Asymptotics**  
@@ -611,8 +610,14 @@ e. **Uniform Convergence and Sequence Behavior**
    - When proving uniform convergence, ensure that the bound obtained is independent of x to establish uniform control.  
    - If using asymptotic behavior (e.g., factorial ratios tending to zero), provide explicit justification rather than just stating the result.  
 
-f. **Logical Flow and Intuition**  
-   - After major steps (e.g., computing a limit, verifying continuity), summarize why the step is necessary and how it advances the proof.  
+f. **Clarify the Use of Key Theorems (e.g., Squeeze Theorem)**  
+   - When using the squeeze theorem, clearly state why both bounding functions tend to the same limit and explicitly apply the theorem in the conclusion.  
+
+g. **Logical Flow and Transitions**  
+   - After major steps (e.g., computing a limit, verifying continuity), summarize why the step was necessary and how it connects to the next part of the proof.  
+   - Ensure smooth logical transitions between different parts of the proof by briefly explaining why one step leads naturally to the next.  
+
+h. **Concluding and Intuitive Explanations**  
    - Conclude with an intuitive explanation of why the result makes sense, possibly connecting it to known theorems or simple examples.  
    - In notes after the proof, highlight potential sources of confusion for students and clarify tricky aspects of the problem.  
 """