Hello, World (but with Lemmas)

Welcome — inline math like $a^2 + b^2 = c^2$ and display:

$\int_0^1 x^2 \, dx = \frac{1}{3}.$

Lemma : Triangle Inequality

For all real numbers $x,y$ : $|x+y| \le |x| + |y|$ .

Proof

Follows from the norm properties (or by squaring and case analysis).

Exercise : Young's inequality (1D)

Show $|ab|\le \tfrac12(a^2+b^2)$ for $a,b\in\mathbb{R}$ .