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?

Hint: I'd rewrite the set as
$$C = \{x\in{\Bbb R}\mid -1/\sqrt 2\leq x\leq 1/\sqrt 2\} = \left[-\frac{1}{\sqrt 2};\frac{1}{\sqrt 2}\right].$$
