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Prove $\lim\limits_{n\to\infty}(a+b+c+…)^n=a^n+b^n+c^n+…$

I have already read a number of proofs on this site of this statement, which goes as follows:
$$\lim_{n\to\infty}\left(\frac{a+b+c+…+z}{z}\right)^n=\frac{a^n+b^n+c^n+…+z^n}{z^n}$$
My problem is that the proof below:
$$\lim_{n\to\infty}\left(a^n+b^n+c^n+…+z^n\right)=\sum_{n=0}^{\infty}\lim_{n\to\infty}\left(a^n+b^n+c^n+…+z^n\right)$$
is not complete. I understand the parts $\lim_{n\to\infty}\left(a^n+b^n+c^n+…+z^n\right)=\sum_{n=0}^{\infty}\left(a^n+b^n+c^n+…+z^n\right)$ and $\sum_{n=0}^{\infty}\left(a^n+b^n+c^n+…+z^n\right)=\lim_{n\to\infty}\left(a^n+b^n+c^n+…+z^n\right)$, but i do not understand why we can say that the n-th element is $\lim_{n\to\infty}\left(a^n+b^n+c^n+…+z^n\right)=0$.
I hope someone can give me an explanation of this question.

The idea is that the limit definition of $\lim\limits_{n\to\infty}\left(a^n+b^n+c^n+…+z^n\right)$ is $\text{limit

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