Implications of convergence in probability

Question

Consider two sequences of real-valued random variables $\{X_n\}_n$, $\{Z_n\}_n$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Suppose that

(1) $Z_n\in o_p(1)$, i.e. $Z_n$ converges in probability to $0$

(2) $X_n+Z_n\geq 0$ $\forall n$

(3) $X_n\leq 0$ $\forall n$

I want to show that $X_n\in o_p(1)$

Any suggestion? I get the intuition but I don't know how to show it formally

Answer 1

Note that \begin{align} \mathbb P\{|X_n|\gt \varepsilon\}&=\mathbb P\{X_n\lt -\varepsilon\} \quad\mbox{ by }(3)\\ &=\mathbb P\{X_n+Z_n\lt Z_n-\varepsilon\}\\ &\leqslant \mathbb P\{0\lt Z_n-\varepsilon\}\quad\mbox{ by }(2)\\ &=\mathbb P\{\varepsilon\lt Z_n\}\\ &\leqslant\mathbb P\{ |Z_n|\gt\varepsilon \} , \end{align} hence by (1), we get the wanted result.