  Extragradient method (EG) (Korpelevich, 1976) is one of the most popular
methods for solving saddle point and variational inequalities problems (VIP).
Despite its long history and significant attention in the optimization
community, there remain important open questions about convergence of EG. In
this paper, we resolve one of such questions and derive the first last-iterate
$O(1/K)$ convergence rate for EG for monotone and Lipschitz VIP without any
additional assumptions on the operator unlike the only known result of this
type (Golowich et al., 2020) that relies on the Lipschitzness of the Jacobian
of the operator. The rate is given in terms of reducing the squared norm of the
operator. Moreover, we establish several results on the (non-)cocoercivity of
the update operators of EG, Optimistic Gradient Method, and Hamiltonian
Gradient Method, when the original operator is monotone and Lipschitz.
