Is there a meaningful sense in which a sequence of PMFs (of a corresponding sequence of real-valued random variables) can uniformly converge to a PDF? Intuitively, it seems like a strange question to consider (and that the answer would be no) since a PMF is a function $f: \mathbb{R} \mapsto [0, 1]$, while a PDF is a function $f: \mathbb{R} \mapsto [0, \infty)$.
However, what about the following case: Say that we have a sequence of standardized random variables, which are denoted by $\left\{X_n\right\}_{n \in \mathbb{N}}$. Denote the corresponding sequence of CDFs by $\left\{F_n(x)\right\}_{n \in \mathbb{N}}$ and let this sequence of CDFs uniformly converge in distribution to $F(x)$, where $F(x)$ is the standard Normal CDF. That is,
$$\text{For all } \epsilon > 0, \text{ there exists an } N \text{ such that } \left\lvert F_n(x) - F(x) \right\rvert < \epsilon \text{ for all } n > N \text{ and for all } x.$$
Also, denote the corresponding sequence of PMFs for the sequence of standardized random variables by $\left\{f_n(x)\right\}_{n \in \mathbb{N}}$ and denote the PDF of the aforementioned limiting standard Normal distribution by $f(x)$. The maximum density of a standard Normal PDF is $\frac{1}{\sqrt{2\pi}} \approx 0.3989$.
Therefore, in this particular case, where the density of the standard Normal has an upper bound of $\frac{1}{\sqrt{2\pi}}$, it seems that it might make sense to consider the conditions under which the sequence of PMFs $\left\{f_n(x)\right\}_{n \in \mathbb{N}}$ uniformly converges to $f(x)$. On the other hand, the statement that a sequence of standardized random variables, $\left\{X_n\right\}_{n \in \mathbb{N}}$, uniformly converges to a standard Normal can usually be equivalently stated as the sequence of centered random variables scaled by $\sqrt{n}$ uniformly converges in distribution to a Normal distribution with mean $0$ and variance equal to $\nu > 0$, where now the Normal density is back to being unbounded.
So my question is: Would it be sensible in the case of a sequence of standardized random variables to give conditions under which the sequence of PMFs uniformly converges to a PDF? And if so, what might those conditions be? Or is convergence of a sequence of PMFs to a PDF a nonsensical thing to consider in general?