If a Stochastic Process has Variance linear with t, how to prove it is not Wide Sense Stationary?

Question

For my study, as a part of a Matlab exercise, the following question is asked:

Using the results of the estimated standard deviations of the random variable $x(k)$ for $k = 10^3; 10^4; 10^5$ conclude whether the random process is wide sense stationary (WSS) or not.

The process we are studying in this exercise is Brownian motion, using the following difference equation:

$x(k)+\beta_1x(k-1)+\beta_2x(k-2)=\beta_3w(k)$

Where $w(k)$ is modeled using samples from the Normal distribution, normalised by the time step $dt$.

Now, from theory, I know that the variance of the Brownian motion (a Wiener process) increases linearly with time:

$Var(x)=at=\sigma_x^2$

So, $\sigma_x=b\sqrt(t)$

This is indeed what I observe, when I plot these values; the function looks like a square root. When I plot the squared values, I get something looking very nicely linear.

I also know that a Wiener process is not WSS from literature (the internetz). Now, it is up to me to prove this from the gained relation between the standard deviation (Variance), and the time.

According to my course reader, 3 criteria are needed for a stochastic process to be WSS:

1. The mean should be constant (time-invariant)
2. Autocorrelation $t1$,$t2$ should only be dependent on the difference between two time-intervals, $t2-t1$
3. $c_x(0)<\infty$, (with $c_x$ the autocovariance function). i.e. The variance is finite.

I think the criterium that needs to be checked is the 3rd one. Maybe the second one can also be used. (The first one is already met, since for this process the mean is zero, and therefore constant in time.)

I was thinking along the lines of:

$\lim_{t\to\infty} at=\infty$

(Apologies if this is not the correct way to write that) However, I'm not sure if this a valid proof, or not. In a real physical system, t will never reach infity, and the Variance will be very large, but not infinite.

Maybe I can relate the Variance in some way to the Autocorrelation, and this way prove that for a variance linear with t, it is not possible to satisfy condition 2?

I appreciate nudges in the good direction more than final answers, if possible.