Proving that the sublevel set of a quadratic function is convex 
Let
$$C := \{ x \in \mathbb{R} : 2x^2 \le 1 \}$$
Prove that $C$ is convex.

I started with the definition: for $ x_1, x_2 \in C$
$$ \lambda x_1 + (1-\lambda)x_2 = \dots $$
but didn't make any progress. How should I approach this problem?

Alternatively, use the fact that the function $f(x)=x^2$ is convex. Then,
$$2(\lambda x_1 + (1-\lambda) x_2)^2 \leq 2\lambda x_1^2 + 2(1-\lambda)x_2^2 \leq \lambda + 1 - \lambda = 1,$$
where the second inequality follows from $x_1,x_2 \in C.$
