Well, this depends on your definition of white noise. This question asks for that definition.
One answer gives:
A white noise process is a random process of random variables that are uncorrelated, have mean zero, and a finite variance. Formally, $X(t)$ is a white noise process if
$E(X(t))=0,E(X(t)^2)=S^2$, and $E(X(t)X(h))=0$ for $t≠h$.
A slightly stronger condition is that they are independent from one another; this is an "independent white noise process."
Under this definition (the first of the two, the weaker one), which is I presume the same definition you have, your reasoning is perfectly correct and white noise is always wide-sense stationary. Note that this definition asks for the variances be finite, what I think you also do since you probably mean a finite number when you write $c_0$.
For the stronger version of the definition given in the same quoted answer, the same applies.
In contrast, the definition that Dilip Sarwate gave in his answer doesn't require the variances to be finite and hence allows for a white noise not to be wide-sense stationary as he explained.
There are probably other definitions for white noise out there. Possibly, in the context of your exam another definition of white noise is assumed than the one in your book and therefore the apparent contradiction.