  We consider an elliptic variational inequality with unilateral constraints in
a Hilbert space $X$ which, under appropriate assumptions on the data, has a
unique solution $u$. We formulate a convergence criterion to the solution $u$,
i.e., we provide necessary and sufficient conditions on a sequence
$\{u_n\}\subset X$ which guarantee the convergence $u_n\to u$ in the space $X$.
Then, we illustrate the use of this criterion to recover well-known convergence
results and well-posedness results in the sense of Tykhonov and Levitin-Polyak.
We also provide two applications of our results, in the study of a heat
transfer problem and an elastic frictionless contact problem, respectively.
