point-wise boundness of random sequence in a sequence space Assume we have a sequence of random elements $\{X_{n}\}_{n\geq 1}$ taking values in sequence space $\ell_{1}$, i.e. for each $n$ one has $X_{n}\in\ell_{1}$.
Next, let us assume that for any finite fixed $k$, the the sequence of random subvertors $(X_{n,i})_{i=1}^{k}\in\mathbb{R}^{k}$ is bounded in probability,
$$
(X_{n,i})_{i=1}^{k} = O_{p}(1)
$$ with respect to the norm of $\mathbb{R}^{k}$.
Does it mean that $\{X_{n}\}_{n\geq 1} = O_{p}(1)$ with respect to the norm of $\ell_{1}$?
PS I am aware that pointwise convergence does not follow the convergence in the norm of the sequence space. Though, I could not figure it out for boundless in probability.

Constant random variables $X_{n,i}=1$ if $i \leq n$ and $0$ if $i >n$  give a counter-example.
