Lemma: An independent increments process $(X\_t)\_{t∈R^+}$ such that $X\_0 = 0$ with Gaussian increments (i.e. for all s < t, $X\_t − X\_s$ has Gaussian law) is a Gaussian process.

Proof:

So what I want to prove is that $(X\_{t\_1},...,X\_{t\_n})$ is a multivariate normal. I see that i can rewrite $X\_{t\_j}=\sum\_{k=1}^j(X\_{t\_k}-X\_{t\_{k-1}})$ therefore i know that each $X\_{t\_j}$ is a sum of independent increment and then is Normal. 

Now, how can I conclude that $(X\_{t\_1},...,X\_{t\_n})$ is multivariate?

Using the definition of multivariate i should show that $$\sum\_{i=1}^na\_iX\_{t\_j}=\sum\_{i=1}^na\_i\sum\_{k=1}^j(X\_{t\_k}-X\_{t\_{k-1}})$$ is normal. But how could i do that? because in this case $\sum\_{i=1}^na\_iX\_{t\_j}$ the $X\_{t\_j}$ seems to me that is not independent among each element, so I cannot conclude that is a sum of independen normal (and from here conclude that is normal)

Please some help, I thought a lot about this problem.