Weakly Convergent Subsequence: Iff-Condition?

Helly's theorem states that we can extract a convergent subsequence from any sequence of distribution functions. Moreover, one can show that the limit of this subsequence is a distribution function (and hence we have weak convergence), if the distribution functions have a tight distribution.

In the lecture it was mentioned, that we would have an iff-condition for being able to extract a convergent subsequence to a distribution function. Does this mean that if the measure is not tight, then the resulting function is not a distribution function, hence there is no weak convergence?

One can try a typical example of non-tight sequence of probability measures on the real line $\mu_n:=\delta_n$. The distribution function of $\mu_n$ is $F_n\colon t\mapsto \mathbf 1_{[n,+\infty)}(t)$. The sequence $(F_n(\cdot))_{n\geqslant 1}$ (as well as all its subsequences) converges pointwise to the null function, which is not a distribution function.

In the general case, if a sequence of probability measures on the real line converges in distribution to a probability measure, then the considered sequence is tight.