Why does this equivalence of sums hold? I have to prove that a function is a pdf. In the master solution they state the below equivalence in the proof. What rules are applied to get from the left side to the right side?
$$C\sum_{k=2}^{\infty} \sum_{j=1}^{k-1}\left(\frac{1}{2}\right)^{k}=C \sum_{j=1}^{\infty} \sum_{k=j+1}^{\infty}\left(\frac{1}{2}\right)^{k}$$

The equivalence can be easily shown by double counting according to the following scheme

indeed

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*$\sum_{k=2}^{\infty} \sum_{j=1}^{k-1}\left(\frac{1}{2}\right)^{k}$ is the sum column by column

*$ \sum_{j=1}^{\infty} \sum_{k=j+1}^{\infty}\left(\frac{1}{2}\right)^{k}$ is the sum row by row

