Erdős Problem #419 — Limit Points of the Factorial Divisor Ratio

Let $\tau(n)$ denote the number of positive divisors of $n$, and define

$$R_n=\frac{\tau((n+1)!)}{\tau(n!)}.$$

What is the set of limit points of $(R_n)_{n\in\mathbb{N}}$ as $n\to\infty$? The claim is that this set is exactly

$$\{1\}\cup\left\{1+\frac1k:k\in\mathbb{N},\ k\geq1\right\}.$$