Fix a probability space $(\Omega,\mathcal A,P)$ and $\mathcal F\subset \mathcal A$ a sub-$\sigma$-algebra of $\mathcal A$. Let $L^0(\mathcal F)$ denote the set of equivalence classes of $\mathcal F$-measurable real random variables $X$ on $(\Omega, \mathcal A,P)$. For $\mathcal S \subset L^0(\mathcal A)$ define
$$\text{span}_{L^0(\mathcal F)} (\mathcal S):=\bigg\{\sum_{i=1}^n Y_iX_i: X_i\in \mathcal S, Y_i\in L^0(\mathcal F), n\in\mathbb N\bigg\}$$
Am asking whether $\text{span}_{L^0(\mathcal F)} (\mathcal S)$ is sequentially closed under convergence in probability.
Consider the following example: $(\Omega,\mathcal A,P):=([0,1],\mathcal B[0,1],\lambda)$ with $\lambda$ the Lebesgue measure on $[0,1]$, and $\mathcal F$ the trivial $\sigma$-algebra on $[0,1]$. Define
$$\text{span}_{L^0(\mathcal F)} (\mathcal S):=\text{span}_{L^0(\mathcal F)} \big\{1_{[0,1-2^{-n}]}: n\in\mathbb N\big\}$$
Then $1_{[0,1-2^{-n}]}$ converges in probability to $1_{[0,1]}$, but $1_{[0,1]}\notin \text{span}_{L^0(\mathcal F)} (\mathcal S)$.
Question: Can this happen if $\mathcal S$ is finite?
Thanks for your help.
EDIT: An attempt for the case $\mathcal F=\mathcal A$ and $\mathcal S=\{X_1,X_2\}$.
Let $(Y_n):=(W^1_nX_1+W^2_nX_2)$ be a sequence in $\text{span}_{L^0(\mathcal F)} (\mathcal S)$, and suppose that $(Y_n)$ converges in probability to $Y\in L^0(\mathcal A)$. There is a subsequence $(Y_{n_k})$ of $(Y_n)$ such that $Y_{n_k}\to Y$ as $k\to\infty$, $P$-almost surely. Let $N\in \mathcal A$ be a null set outside of which $Y_{n_k}\to Y$ pointwise. Define
$$W_1(\omega):=\begin{cases} \frac{Y(\omega)}{2X_1(\omega)}& \text{if } \omega\in \{X_1\neq 0\}\cap\{X_2\neq 0\} \\ \frac{Y(\omega)}{X_1(\omega)}& \text{if } \omega\in \{X_1\neq 0\}\cap\{X_2=0\} \\ 0 & \text{otherwise} \end{cases}$$
$$W_2(\omega):=\begin{cases} \frac{Y(\omega)}{2X_2(\omega)}& \text{if } \omega\in \{X_1\neq 0\}\cap\{X_2\neq 0\} \\ \frac{Y(\omega)}{X_2(\omega)}& \text{if } \omega\in \{X_2\neq 0\}\cap\{X_1=0\} \\ 0 & \text{otherwise} \end{cases}$$
Then $W_1,W_2 \in L^0(\mathcal A)$ and $Y_{n_k}\to Y=W_1 X_1 +W_2 X_2 \in \text{span}_{L^0(\mathcal F)} (\mathcal S) $ outside of $N$, i.e. $Y=W_1 X_1 +W_2 X_2$ $P$-almost surely. So $\text{span}_{L^0(\mathcal F)} (\mathcal S)$ is sequentially closed.
Is this correct?