Limit points of $\left\{\frac{1}{2^m}+\frac{1}{3^n}+\frac{1}{5^p}\ : \ m,n,p\in\mathbb N\right\}$

I intend to find the limit points of the set $\mathcal S = \left\{\frac{1}{2^m}+\frac{1}{3^n}+\frac{1}{5^p}\ : \ m,n,p\in\mathbb N\right\}$.

These are the ones I could think of:

• $S_1 = \left\{\frac{1}{2^m}+\frac{1}{3^n}\ : \ m,n\in\mathbb N\right\}$
• $S_2 = \left\{\frac{1}{2^m}+\frac{1}{5^p}\ : \ m,p\in\mathbb N\right\}$
• $S_3 = \left\{\frac{1}{3^n}+\frac{1}{5^p}\ : \ n,p\in\mathbb N\right\}$
• $S_4 = \left\{\frac{1}{k^n}\ : \ n\in\mathbb N, \ k\in\{2,3,5\}\right\}$
• $S_5=\{0\}$

So the set of limit points of $\mathcal S$ is $\bigcup\limits_i S_i$.

Am I missing anything or that's it? In an exam setting, how do I write this more formally i.e., show that there are no limit points other than these?