Helly's selection principle(in the context of Probability Theory and not Functional Analysis) gives that if $F_{n}$ be a sequence of CDF's then there exists a subsequence $F_{n_{k}}$ such that $F_{n_{k}}(x)\to F(x)$ for all $x$.
But $F$ need not necessarily be a cdf unless the subsequence $F_{n_{k}}$ is tight.
Now, I can produce examples of $F_{n}$ where the sequence is not tight and hence the limiting distribution is not a cdf.
But can we produce an example where $F_{n}(x)$ and $F(x)$ are continuous functions but $F$ is not necessarily a cdf? (i.e. $F(\infty)<1$) ?
Say for example a distribution with a pdf like $\frac{1}{2}\exp(-x)$ . Can there exist a sequence of cdf's (continuous or non-continuous) $F_{n}$ that converge to $F$ which is a distribution function such that $F(x)=\int_{-\infty}^{x}\frac{1}{2}\exp(-x)\mathbf{1}_{(0,\infty)}\,dx$ ?
Answer for the first part: $F_n(x)=0$ for $x<n$, $F_n(x)=(x-n)$ for $n\leq x<n+1$ and $F_n(x)=1$ for $x\geq n+1$. Take $F(x)=0$ for all $x$.
Second part: Let $G$ be any sub-probability distribution. ($G(\infty) <1$). Let $F_n$ be as above and consider $t_nF_n+(1-\frac 1 n) G$ where $t_n$ is defined by $t_n+(1-\frac 1n) G(\infty)=1$. This is sequence of genuine probability distrbutions converging pointwise to $G(x)$. Note also that $t_nF_n+(1-\frac 1 n) G$ is continuous if $G$ is continuous.