Update app.py

app.py

@@ -585,20 +585,38 @@ STRICT REQUIREMENTS:
 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:
-    - Give detailed reasoning for each step and justify every step.
-    - In delta-epsilon proofs, explain clearly why a given choice of delta will work.
-    - Ensure that every bounding argument is explicitly justified
-    - Connect each step with a rationale
-    - When using supremum/infimum, explain why it behaves as expected under limits, differentiation, or integration. Provide explicit justification that the supremum argument does not introduce discontinuities.
-    - If you conclude certain terms vanish in a limit, clearly justify why
-    - If you state that a function has a certain property, such as being Riemann integrable or compact or uniformly continuous for example, clearly explain why
-    - When using limit substitutions, explicitly justify why the transformation is valid and preserves the limit structure.
-    - If verifying differentiability, explicitly state why continuity at that point is necessary and how it connects to the derivative’s existence."
-    - When proving differentiability or continuity, confirm whether the function behaves symmetrically for positive and negative values approaching the given point. If necessary, compute left-hand and right-hand derivatives."
-    - After completing a major step (e.g., computing a limit or verifying continuity), briefly explain why that step matters in the overall proof structure."
-    - Conclude with a brief intuitive explanation of why the result makes sense, possibly by connecting it to known theorems or simple examples.
-    - In notes after the proof, if you observe aspects of the problem that might confuse students, address them."""
-
+   ### Real Analysis Proof Guidelines  
+For Real Analysis proofs, follow these principles to ensure clarity, rigor, and logical completeness:  
+
+a. **Justify Every Step**  
+   - Provide detailed reasoning for each step and explicitly justify every bounding argument, inequality, or limit claim.  
+   - If concluding that terms vanish in a limit, clearly explain why.  
+   - When using supremum/infimum, justify its behavior under limits, differentiation, or integration, ensuring it does not introduce discontinuities.  
+
+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 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.  
+   - When assuming an integral is finite, provide justification based on function class properties (e.g., Riemann integrability implies boundedness).  
+
+d. **Inequalities and Asymptotics**  
+   - When using inequalities (e.g., Hölder’s, Jensen’s), explain why they apply and what function properties make them relevant.  
+   - If using factorial ratios or infinite series sums, explicitly state their rate of convergence and reference known bounds (e.g., Stirling’s approximation).  
+
+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.  
+   - 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.  
+"""
+        
 #Consider
    #When writing SymPy code:
    #- Use FiniteSet(1, 2, 3) instead of Set([1, 2, 3]) for finite sets