Big O notation preserved under convex functions?

Question

Suppose that the random variable $X_T$ is $O_p(1)$ as $T \rightarrow \infty$, i.e. $\forall \epsilon>0$, $\exists M_\epsilon>0$ such that $\mathbb{P}(X_T>M_\epsilon)<\epsilon$ $\forall T$.

Does this imply that the random variable $\max\{0,X_T \}$ is $O_p(1)$ as $T \rightarrow \infty$?

Answer 1

For a real number $x$, $\max\{0,x\}\leqslant |x|$, hence $$\{\max\{0,X_T\}>M\}\subset |X_T|>M\}$$ and the conclusion follows.