Once constructed the Brownian Motion following the Levy-Ciesielski procedure for $t \in [0,1]$ I am aiming to prove the existence of this process for times in $[0, +\infty)$. This is, proving the existence of a stochastic process $\{B_t; t \in [0, + \infty)\}$ such that is a Brownian Motion. Given a countably family of independent Brownian Motions in $[0,1]$, $\{B^{(k)}_t; t \in [0,1]\}_{k=1}^\infty$ we can recursively build $$ B_t := \sum^n_{j=1} W^{(j)}_1 + W^{(n+1)}_{t - n} $$ for any $t \in [n,n+1]$ and prove this is a Brownian Motion following the classical definition ($B_0 \equiv 0$, $B_t - B_s \sim N(0,t-s)$ and independent increments, ($W_{t_1}),(W_{t_2}-W_{t_1}),(W_{t_3}-W_{t_2}),...,(W_{t_m}-W_{t_{m-1}})$ for any $0<t_1<...<t_m< \infty$)
I am struggling trying to prove the last point since the increments overlap due to construction and using characteristic functions leads to lines and lines of aparently not independent random variables.
Moreover, every book I've searched on leaves this result as a "clearly obvious" corollary. As a reference on this topic I am following "An Introduction to Stochastic Differential Equations" from Lawerence C. Evans.
Does anyone know how should I complete this proof? Any tips? Useful results or references?
Thanks in advance.