Suppose I have two iid sequences of random variables $\{X_t\}_{t\in\mathbb{N}}$ and $\{Y_t\}_{t\in\mathbb{N}}$, both absolutely continuous with full support. I know that with probability one, for any $M>0$ we have that $|X_t|<M$ for infinitely many $t$ and thus $$ P(\liminf_{t\rightarrow\infty} |X_t| = \infty) = 0. $$ The same is true for $\{Y_t\}$.
Question: Do we have $$ P(\liminf_{t\rightarrow\infty} \,\, \max\{|X_t|,|Y_t|\} = \infty) = 0? $$ That is, do we have with probability one that there exists a $M>0$ such that $|X_t|<M$ and $|Y_t|<M$ at the same time, for infinitely many $t$. The answer is unclear to me, because we cannot assume that $\{(X_t,Y_t)\}$ is an iid sequence in $\mathbb{R}^2$.