Prove convexity of squared Euclidean norm

Question

I need to prove that the square of the Euclidean norm is convex, so:

$||\theta x+(1-\theta)y||^2\leq\theta||x||^2+(1-\theta)||y||^2$.

Can I use the triangular inequality (if yes, how?) or should I use something else?

Answer 1

Notice that $f(x)=x^2$ is convex ($2=f''(x)>0$.) Thus it is convex, which means that it satisfies $f\big(\theta\|x\|+(1-\theta)\|y\|\big)\leq \theta f(\|x\|)+(1-\theta)f(\|y\|)$ for $\theta\in [0, 1]$.