Say that we have a sequence of discrete random variables, $\left\{X_n\right\}_{n \in \mathbb{N}}$, which converges to a random variable, $X$, with a continuous distribution, e.g., the Normal (Gaussian) distribution.
Denote the corresponding sequence of PMFs for $\left\{X_n\right\}_{n \in \mathbb{N}}$ by $\left\{g_n\right\}_{n \in \mathbb{N}}$. Is this sequence of PMFs, $\left\{g_n\right\}_{n \in \mathbb{N}}$, uniformly bounded?
Based on the definition I have seen for uniform boundedness, the sequence of PMFs, $\left\{g_n\right\}_{n \in \mathbb{N}}$, would be uniformly bounded when
$$\sup_n \left\lvert g_n(x) \right\rvert \leq M(x) < \infty \text{ for each } x \in \mathbb{R}.$$
Even if the limiting random variable has a continuous distribution, it would seem that uniform boundedness of the sequence of PMFs trivially holds: For each $n = 1, 2, 3, \dots$ in the sequence, $\left\{g_n\right\}_{n \in \mathbb{N}}$, the largest possible value of $g_n(x)$ is $1$ since $g_n$ is a probability mass function. Hence, wouldn't letting $M(x) = M \geq 1$ for every $n \in \mathbb{N}$ ensure that $\left\{g_n\right\}_{n \in \mathbb{N}}$ satisfies uniform boundedness as defined above? Or is my reasoning incorrect?
Many thanks to all for any help.