<s>[INST] Given $a^2+b^2=2$ prove $a+b\le2$ 
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*Given $a^2+b^2=2$ prove $a+b\le2$

*Given $a+b=2$ prove $a^4+b^4\ge2$
I was trying to prove these using the fact that we know $a^2+b^2\ge2ab$ but not sure where to start.
 [/INST] Courtsey : $\rightarrow$ Cauchy Schwarz
$$(a+b)^2\le(1+1)(a^2+b^2)\\\implies a+b\le |a+b|\le 2$$
$$$$Now by the fact that $a+b=2$ and $a^2+b^2\ge 2ab$, we deduce that $2(a^2+b^2)\ge 4\implies a^2+b^2\ge 2$ $$(a^2+b^2)^2\leq2(a^4+b^4)\\ \implies a^4+b^4\ge 2$$
BINGO!
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