Consider the sequences of real-valued random variables $\{X_n\}_{\forall n \in \mathbb{N}}, \{Y_n\}_{\forall n \in \mathbb{N}}$. A book that I'm reading makes the following claims whose interpretation is not clear to me.
Case 1: $X_n\leq Y_n+o_p(1)$, i.e. $X_n$ converges to $Y_n$.
Case 2: $X_n\geq Y_n+o_p(1)$, i.e. $X_n$ converges to $Y_n$.
I guess that $o_p(1)$ is a sequence converging in probability to zero as $n\rightarrow \infty$.
I would like your help to clarify firstly what does it mean $X_n\leq Y_n+o_p(1)$ (and, symmetrically, $X_n\geq Y_n+o_p(1)$), why they imply convergence, and which type of convergence.