Does the set $A = \{1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} : n\in \mathbb{N}\}$ have any limit points?

My intuition is that it does not.

No element in $A$ is a limit point, since for any element $x$ in $A$ there is an $\epsilon$-neighborhood of $x$ intersecting $A$ only at $x$. It also seems that I can prove using the density of $\mathbb{Q}$ in $\mathbb{R}$ that any number not in $A$ has some $\epsilon$-neighborhood that intersects $A$ nowhere.

Yet, I doubt that this method will work, because I could do the same thing for $\{1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} : n\in \mathbb{N}\}$, even though this set definitely has a limit point, namely Euler's Sum.