My definition of bounded in probabilty is following:
A random sequence $ \{ x_t : t \in \mathbb{Z} \} $ is said to be bounded in probability, if $$ \lim_{c \rightarrow \infty} \sup_{t \in \mathbb{Z}} P( |x_t|>c) =0. $$ In a paper of A. Klivecka regarding the random GARCH it is said right after this definition:
Clearly, any strictly stationary sequence is bounded in probability. Note that any sequence $\{ x_t : t \in \mathbb{Z} \}$ is bounded in probability if $\sup_{t \in \mathbb{Z}} \mathbb{E}|x_t|^p < \infty $ for some $p>0$ (by Chebychevs inequality).
Why are these staments true? Some more details would be really nice.
If $\{x_{t} \, \mid \, t \in \mathbb{Z}\}$ is strictly stationary, then $\mathbb{P}\{|x_{t}| \geq c\} = \mathbb{P}\{|x_{1}| \geq c\}$ independently of $t$. Moreover, by Chebyshev's inequality, \begin{equation*} \mathbb{P}\{|x_{1}| \geq c\} \leq \frac{\mathbb{E}(|x_{1}|^{p})}{c^{p}}. \end{equation*} Therefore, \begin{equation*} \limsup_{c \to \infty} \sup\{ \mathbb{P}\{|x_{t}| \geq c\} \, \mid \, t \in \mathbb{Z}\} = \lim_{c \to \infty} \mathbb{P}\{|x_{1}| \geq c\} \leq \lim_{c \to \infty} \frac{\mathbb{E}(|x_{1}|^{p})}{c^{p}} = 0. \end{equation*}