In Schilling, Partzsch, referring to Levy-Ciesielski construction of Brownian Motion, I read that:
[...] idea is to write the paths $[0,1]\ni t \mapsto B_t(\omega)$ for almost every $\omega$ as a random series with respect to a complete orthonormal system (ONS) in the Hilbert space $L^2(dt)=L^2([0,1]\text{, }dt)$ with canonical scalar product $\langle f,g\rangle_{L^2}=\displaystyle{\int_{0}^1f(t)g(t)dt}$. Assume that $(\phi_n)_{n\geq0}$ is any complete Orthonormal System and let $(G_n)_{n\geq0}$ be a sequence of real-valued iid Gaussian $\mathbb{N}(0,1)$ - random variables on the probability space $\left(\Omega,\mathcal{A},\mathbb{P}\right)$. Set:
$$W_N(t):=\sum_{n=0}^{N-1}G_n\langle\mathbb{1}_{[0,t)},\phi_n\rangle_{L^2}\tag{1}$$
$$=\sum_{n=0}^{N-1}G_n\displaystyle{\int}_0^t\phi_n(s)ds$$Lemma. The limit $W(t):=\lim_{N\to\infty}W_N(t)$ exists for every $t\in[0,1]$ in $L^2(\mathbb{P})$ [...]
Proof. $\color{red}{\text{Using }}$ the independence of the $G_n\sim\mathbb{N}(0,1)$ and $\color{red}{\text{Parseval's identity}}$, we get for every $t\in[0,1]$
$$\mathbb{E}\left(W_N(t)^2\right)=\mathbb{E}\bigg[\sum_{m,n=0}^{N-1}G_nG_m\langle1_{[0,t)},\phi_m\rangle_{L^2}\langle1_{[0,t)},\phi_n\rangle_{L^2}\bigg]\tag{2}$$
$$=\sum_{m,n=1}^{N-1}\mathbb{E}(G_nG_m)\langle1_{[0,t)},\phi_m\rangle_{L^2}\langle1_{[0,t)},\phi_n\rangle_{L^2}$$
$$=\sum_{n=1}^{N-1}\langle1_{[0,t)},\phi_n\rangle_{L^2}^2\underset{N\to\infty}{\longrightarrow}\langle 1_{[0,t)},1_{[0,t)}\rangle_{L^2}=t$$ $\color{red}{\text{This shows that }W(t)=L^2\text{-}\lim_{N\to\infty}W_N(t)\text{ exists}}$.
I have two doubts regarding the above parts in $\color{red}{\text{red}}$:
- I guess that Parseval's identity is applied to $\sum_{n=1}^{N-1}\langle1_{[0,t)},\phi_n\rangle_{L^2}^2$ in $(2)$ before taking the limit as $N$ goes to $\infty$. But how does it happen? Could you please explicit such an application of the identity?;
- Why does it hold true that $$\lim_{N\to\infty}\mathbb{E}\left(W_N(t)^2\right)=t\implies W(t)=L^2\text{-}\lim_{N\to\infty}W_N(t)\text{ exists}$$ ? Is this due to some basic probability theory result?
Hint
Parseval identity is used in the last step, i.e. for $$\sum_{n=0}^{\infty }\left<\boldsymbol 1_{[0,t]},\varphi _n\right>^2_{L^2}=\|\boldsymbol 1_{[0,t]}\|^2_{L^2}=\left<\boldsymbol 1_{[0,t]}, \boldsymbol 1_{[0,t]}\right>_{L^2}.$$
For your other question, the fact that $$\lim_{N\to \infty }\mathbb E[W_N(t)^2]=t,$$ implies that $(W_N(t))_N$ is an $L^2(\mathbb P)-$Cauchy sequence.
Edit
One can prove that $$\mathbb E[W_n(t)W_m(t)]=\mathbb E[W_{n\wedge m}(t)^2],$$ and thus
\begin{align} \mathbb E[(W_n(t)-W_m(t))^2]&=\underbrace{\mathbb E[W_n(t)^2]+\mathbb E[W_m(t)^2]-2\mathbb E[W_{n\wedge m}(t)^2]}_{\underset{n,m\to \infty }{\longrightarrow }0}. \end{align}