Let $X_{1},X_{2},\cdots$ be real random variables identically distributed. We consider the sequence $m_{n} := Med(X_{n})$, where $Med(\cdot)$ denotes the Median of a random variable. My question is whether or not $m_{n}$ is a bounded sequence, that is, if there exists $M\in\mathbb{R}$ such that $m_{n}\leq M$ for all $n\in\mathbb{N}$.
My intuition says yes, it is difficult to imagine an example that does not satisfy this.
Thanks in advance.