Let's define a probability space $(\Omega$, $\mathcal{F}$, $\mathbb{P})$ and a sequence of random variables $(X_n)_{n\geq0}$ defined on it.
If I assume that the limit of such a sequence of random variables exists, that is \begin{equation*} \lim\limits_{n\rightarrow\infty}X_n = \liminf\limits_{n\rightarrow\infty} X_n = \limsup\limits_{n\rightarrow\infty} X_n = X \end{equation*} considering that $\liminf\limits_{n\rightarrow\infty}X_n = \lim\limits_{n\rightarrow\infty}\bigg(\inf\limits_{k\geq n}X_n\bigg)$ and $\limsup\limits_{n\rightarrow\infty}X_n = \lim\limits_{n\rightarrow\infty}\bigg(\sup\limits_{k\geq n}X_n\bigg)$, could I say that the above equation corresponds to stating that: \begin{equation*} \lim\limits_{n\rightarrow\infty}X_n = \inf\limits_{n\rightarrow\infty} X_n = \sup\limits_{n\rightarrow\infty} X_n = X \end{equation*} ?
I think that it is not possible, but I am searching for confirmation/rejection of my consideration.