Inequality. $a^2+b^2+c^2 \geq a+b+c$

Using Cauchy-Schwarz inequality we get $$ a+b+c=a\cdot 1+b\cdot 1+c\cdot 1+\leq\sqrt{a^2+b^2+c^2}\sqrt{1^2+1^2+1^2}\tag{1} $$ From AM-GM we obtain $a^2+b^2+c^2\geq 3\sqrt[3]{a^2b^2c^2}=3$, so $$ \sqrt{3}\leq\sqrt{a^2+b^2+c^2}\tag{2} $$ From $(1)$ and $(2)$ it follows $$ a+b+c\leq\sqrt{a^2+b^2+c^2}\sqrt{3}\leq\sqrt{a^2+b^2+c^2}\sqrt{a^2+b^2+c^2}=a^2+b^2+c^2 $$

Let's solve it in an elementary way and start from the fact that: $$a^2 \ge 2a -1 \tag1$$ $$b^2 \ge 2b-1 \tag2$$ $$c^2 \ge 2c-1 \tag3$$

Then add up $(1)$ $(2)$ $(3)$ and get: $$a^2+b^2+c^2 \ge a+b+c +a+b+c -3 \tag4$$ By AM-GM we have $$\frac{a+b+c}{3} \ge (abc)^\frac{1}{3}=1 $$ $$ a+b+c \ge 3 \tag5 $$ Finally, from $(4)$ and $(5)$ we obtain the required inequality: $$a^2+b^2+c^2 \ge a+b+c +a+b+c -3 \ge a+b+c $$


As A.M. of any set of positive number ≥ G.M,

$\frac{a+b+c}{3}≥ (abc)^{\frac{1}{3}}$

Now, $ 3(a^2+b^2+c^2)-(a+b+c)^2= (a-b)^2+(b-c)^2+(c-a)^2≥ 0$

=>$ a^2+b^2+c^2≥ \frac{(a+b+c)^2}{3}$

=>$ a^2+b^2+c^2≥(a+b+c)(\frac{a+b+c}{3})≥a+b+c$ as $\frac{a+b+c}{3}≥1$