Which type of convergence implies that in each iteration we are closer?

Question

Please forgive me if it is a simple answer or if I didn't get something completely, because I don't understand various types of convergence clearly. In a theorem, I can prove convergence in probability that say $\hat{\theta}_n \rightarrow \theta$ as $n\rightarrow \infty$. But what I need is the following \begin{eqnarray} \vert \hat{\theta}_{n}-\theta \vert \lt \vert \hat{\theta}_{n_0}-\theta \vert,\qquad \forall n \geq n_0, \text{possibly with probability} \quad 1-\epsilon \end{eqnarray} What kind of convergence, such as convergence in probability, convergence with probability one or other types of convergence implies this inequality and how to reach from that type of convergence to the above inequality?