This might be an easy question, but I just can't figure out the steps in this equation in a proof.
$ S_n(g,t) $ is the n-th partial Sum : $ \sum_{-n}^n \hat{g} (j) e^{ijt} $ And also here you need to use that you can write $ \hat{f}(j)= \Omega_j \hat{g}(j) $
For me it woud be: $$S_n(g,t)-S_m(g,t)= \sum_{-n}^n \hat{g}(j)e^{ijt}- \sum_{-m}^m \hat{g}(j)e^{ijt}$$ $$= \sum_{-n}^n \hat{g}(j)e^{ijt} \Omega_j \frac{1}{ \Omega_j}- \sum_{-m}^m \hat{g}(j)e^{ijt} \Omega_j \frac{1}{ \Omega_j}= \sum_{-n}^n \hat{f}(j)e^{ijt} \Omega^{-1}- \sum_{-m}^m \hat{f}(j)e^{ijt} \Omega^{-1} $$
Also know about the cauchy criteria for sequences..but this just doesn't add up for me.. I appreciate any help!!