Can someone please help with the following problem. I need to write the following sum in Cantor normal form:
$$\sum_{i ∈ ω\cdot2} \sum_{j ∈ i} (i+j)$$
The result I'm getting is $$ω^2 + w,$$ so I would like to check if it's correct.
Thank you very much in advance!
Edit to add:
My work: $$\sum_{i ∈ ω\cdot2} \sum_{j ∈ i} (i+j) = \sum_{i ∈ ω} \sum_{j ∈ i} (i+j) + \sum_{i ∈ ω} \sum_{j ∈ i} (i+ω+j) = \sum_{i ∈ ω} \sum_{i∈j} (i+j+i+ω+j) = \sum_{i∈ ω} ( (i+0+i+ω+0)+(i+1+i+ω+1)+...+(i+i+i+ω+i)) = \\ sup_{n} \frac{2n(2n-1)}{2} + n\cdotω = ω^2 + ω$$
I'm not sure about the part that involves $$\frac{2n(2n-1)}{2}$$ I've concluded that the supremum for that is $$ω$$ but I'm not sure if that's correct