<s>[INST] Does $a \leq b + c $ imply $a^2 \leq (b+c)^2 + (b-c)^2$? Givens
$$
a \leq b + c 
$$
or 
$$
a^2 \leq b^2 + c^2+2bc 
$$
Can we prove that?: 
$$
a^2 \leq (b+c)^2 + (b-c)^2 = 2b^2 + 2c^2
$$
 [/INST] If
$$a^2 \leq b^2+c^2 + 2bc= (b+c)^2,$$
then 
$$a^2 \leq (b+c)^2 + (b-c)^2$$
as $(b-c)^2 \geq 0$.
</s>