Question on the proof of existence of polynomials

[yth2012]

Not sure if this repo is still monitored or responded to but I guess i will try my luck asking here!
I am reading the part where readers are encouraged to work out the existence of polynomials by examinating

the book said "clearly the constructed polynomial f(x) has degree n because each term has degree n. Now I can understand each term has degree n, but I don't know how from "each term has degree n" leads to "f(x) has degree n", because I think that since this is product, we multiply each term so the degree will be n*(total number of term)?

[j2kun]

The inner-most fraction has degree 1, and the product has n such degree-1 fractions. So each of the products has degree n. Then you sum n+1 of those degree-n terms, and summing does not increase the degree (it can decrease it, maybe).