In my lecture notes I was trying to understand the proof of the following theorem.The notes say that its easy to verify that $H=\{X \in L^2(\Omega, \mathcal{F}_{\infty}^B,\mathbb{P}), X=a+ \int_{0}^{\infty}\phi_s dB_s, \phi \in \Lambda^2 \}$ is closed subspace of $L^2(\Omega, \mathcal{F}_{\infty}^B,\mathbb{P})$. I am quite ashamed to say that I cannot prove this seemingly elementary fact and I was wondering ig someone could help me here?
I need to show that if $(x_n) \in H$ such that it converges to some $x$, then $x \in H$. If $(x_n) \subseteq H$ converges then its a cauchy sequence in $H$ and therefore in $L^2(\Omega, \mathcal{F}_{\infty}^B,\mathbb{P})$ and since its complete it lies in $L^2(\Omega, \mathcal{F}_{\infty}^B,\mathbb{P})$ but I couldn't not show that it lies in $H$ too. Any hints would be appreciated.
Thanks
