I am getting confused with the following set of notes http://www.stat.umn.edu/geyer/8112/notes/metric.pdf.
On page 10, the author states Prokhorovs theorem as follows: Let $X_1,X_2,\dots$ be a sequence of random elements of a Polish space. If the sequence is tight, then every subsequence contains a weakly convergent subsubsequence. Conversely, if the sequence weakly converges, then it is tight.
In the next section, he mentions the subsequence principle which says that $X_n$ converges weakly to $X$ if and only if for every subsequence of $X_n$, we can extract a further subsequence which weakly converges to $X$.
However, does this then not mean that if $(X_n)$ are random elements in a Polish space, then tightness is equivalent to weak convergence?
This seems fishy to me. Could somebody please expand on this?