<s>[INST] How to show that $(a+b)^p\le 2^p (a^p+b^p)$ If I may ask, how can we derive that $$(a+b)^p\le 2^p (a^p+b^p)$$ where $a,b,p\ge 0$ is an integer? 
 [/INST] By induction.
Assume without loss of generalization that $A \leq B$
Note that $A^{n-1}(B-A) \leq B^{n-1}(B-A) \Rightarrow A^{n-1}B - A^n \leq B^n - AB^{n-1} \Rightarrow AB^{n-1} + A^{n-1}B) \leq A^n + B^n$
For $p = 1$ this is obvious. Supose true for all $p < n$.
$(A+B)^n = (A+B)^{n-1}(A+B) \leq 2^{n-1}(A^{n-1}+B^{n-1})(A+B) = 2^{n-1}(A^n + B^n + AB^{n-1} + A^{n-1}B) \leq 2^{n-1}(A^n + B^n + A^n + B^n) = 2^n(A^n + B^n)$
</s>