TITLE: Prove the scaling property of a Brownian motion.
QUESTION [5 upvotes]: I have to prove that $X_t:=c^{-1/2}W_{ct}$, $t\ge0$, where $c>0$ is a constant is a Wiener process.
My attempt:
1) $X_0=c^{-1/2}W_0=0$
2) We know that $(W_t)$ has continuous trajectories. It implies that $(X_t)$ has also continuous trajectories since continuous function multiplied by a constant and with scaled argument is still continuous. Right?
3) Independence of increments. We know that $W_i-W_j$ and $W_k-W_l$ are independent for all $i, j, k, l\in\mathbb{R}^+$. Now, $X_i-X_j=c^{-1/2}(W_{ci}-W_{cj})$ nad $X_k-X_l=c^{-1/2}(W_{ck}-W_{cl})$ so $(X_t)$ has independent increments.
4) $X_{s+\epsilon}-X_s=c^{-1/2}(W_{c(s+\epsilon)}-W_{cs})$ ~ $N(0,\epsilon)$

REPLY [3 votes]: yes, an alternative solution is observing/proving $X_t$ is a Gaussian process and its mean and covariance agrees with the mean and covariance of a Brownian motion, so they must have the same distribution. 
here are two proofs I did for other things to show they are Gaussian.
Brownian Bridge as a Gaussian Process
https://stats.stackexchange.com/questions/78087/i-want-to-show-e-alpha-tbe2-alpha-t-is-a-gaussian-process-and-i-find/81010#81010
$X_t$ is Guassian if $\sum\limits_{i=1}^n \lambda_i X_{t_i}$ is a normally distributed with all value of $\lambda_i\in\mathbb{R}$ and $t_i\geq 0$.