The following question is probably rather naive, so I apologize beforehand.
In class we defined the trivial metric $d$ as follows:
$$d(x,y) = \begin{cases} 0 &\mbox{if } x = y \\ 1 & \mbox{if } x \ne y \end{cases} $$
It was said, that in any space endowed with the trivial metric every sequence converges.
I do not understand why this is true.
If I am not mistaken convergence in this metric requires that a sequence has to be eventually constant; i.e. the values of the sequence have to be constant after a certain index. But this contradicts the claim that every sequence converges.
What am I doing wrong?
You are correct, and the source you cited isn't. In fact, if a sequence is not constant eventually, then it can't be Cauchy because $\limsup\limits_{n\to\infty}d(x_n,x_{n+1})=1$.