<s>[INST] Infinite primes proof There is a proof for infinite prime numbers that i don't understand.

right in the middle of the proof:
"since every such $m$ can be written in a unique way as a product of the form:
$\prod_{p\leqslant x}p^{k_p}$. we see that the last sum is equal to: $\prod_{\binom{p\leqslant x}{p\in \mathbb{P}}}(\sum_{k\leqslant 0}\frac{1}{p^k})$.
I don't see that. can anyone can explain this step to me?
 [/INST] Suppose the prime numbers not exceeding $x$ are $2$, $3$, and $5$.  Then
\begin{align}
& \prod_{\begin{smallmatrix} p\in\mathbb P \\  p\le x \end{smallmatrix}} \sum_{k\ge 0} \frac 1 {p^k} \\[10pt] = {} & \left( 1 + \frac 1 2 + \frac 1 4 + \frac 1 8 + \cdots+ \frac 1 {2^k} + \cdots \right) \left( 1 + \frac 1 3 + \frac 1 9 + \frac 1 {27} + \cdots+ \frac 1 {3^k} + \cdots \right) \times {} \\
{} & {} \times \left( 1 + \frac 1 5 + \frac 1 {25} + \frac 1 {125} + \cdots+ \frac 1 {5^k} + \cdots \right) \tag a \\[12pt]
= {} & 1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \frac 1 5 + \frac 1 6 + \frac 1 8 + \frac 1 9 + \frac 1 {10} + \frac 1 {12} + \frac 1 {15} + \frac 1 {16} \\[6pt]
{} & {} + \frac 1 {18} + \frac 1 {20} + \frac 1 {24} + \frac 1 {25} + \frac 1 {30} + \frac 1 {32} + \frac 1 {36} + \cdots
\end{align}
In this last sum we exclude $1/7$, $1/11$, $1/13$, $1/14$, etc. and include only numbers $1/m$ where $m$ has no prime factors other than $2$, $3$, and $5$.  This last series must converge to a finite number because all three of the series in the line labeled (a) converge to finite numbers.
By contrast, the harmonic series $\displaystyle\sum_{m=1}^\infty \frac 1 m$ diverges to $+\infty$.
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