Littlewood's conjecture

Prove that every pair of real numbers (x,y) satisfies \(\liminf_{q\to\infty} q\,\|qx\|\,\|qy\|=0\), thereby resolving Littlewood's conjecture.

Background

The paper recalls Littlewood's conjecture, which asserts that multiplicative Diophantine approximation in two dimensions cannot remain uniformly bounded away from zero. Although almost every pair satisfies the conjecture and the exceptional counterexample set is known to have Hausdorff dimension zero, the conjecture itself remains unresolved.