TITLE: Why is $\lim\limits_{n\to\infty } 1=0$ incorrect?
QUESTION [6 upvotes]: $$
\lim_{n\to\infty } 1
=\lim_{n\to\infty }\frac{n}{n}
=\lim_{n\to\infty }\frac{\overbrace{1+1+\ldots+1}^{n \text{ times}}}{n}
=\lim_{n\to\infty }\frac{1}{n} + \lim_{n\to\infty }\frac{1}{n} + \ldots
=0.
$$
Clearly this is incorrect, but why?

REPLY [2 votes]: Unlike what Lucian said, it is possible to give a precise meaning (think measure theory) to an infinite sum of limits, and the answer can be 0 under suitable conventions. However, the problem was in the step of splitting the limit. There is a theorem that shows that you can split a limit of a sum of two terms into the sum of their limits provided they exist. That can never result in an infinite sum.