What's wrong with this reasoning that $\frac{\infty}{\infty}=0$?

$$\frac{n}{\infty} + \frac{n}{\infty} +\dots = \frac{\infty}{\infty}$$

You can always break up $\infty/\infty$ into the left hand side, where n is an arbitrary number. However, on the left hand side $\frac{n}{\infty}$ is always equal to $0$. Thus $\frac{\infty}{\infty}$ should always equal $0$.

How do you know $0+0+0+0+...... = 0$? If you think about it $0 + 0 + 0 +..... = 0\times\infty = 0\times 1/0 = 1$. (or any other number). We clearly are dealing with a value on which standard arithmetic doesn't apply. (Hence "infinity is not a number".)

So the question becomes what does apply and how do we deal with this? And that is not an easy/simple question. It's not hard... but it's not simple. Bottom line, finite rules of arithmetic do not always apply, and such instincts lead to common traps.

The existing answers are very good; let me give yet another one.

Consider the counter-argument that ${\infty\over\infty}=\infty$: We have $\infty=2^\infty$, so $${\infty\over\infty}={\infty\over 2^\infty}={\infty\over 2}\cdot {1\over 2}\cdot {1\over 2}\cdot {1\over 2}\cdot...$$ But ${\infty\over 2}$ is infinity, and stays infinity no matter how many times we divide it by $2$. So ${\infty\over \infty}=\infty$.

This is exactly the same "shape" as the argument you give, but yields the opposite answer; so something must be wrong with this kind of argument.

The problem lies in the "number" $\infty\over\infty$. It hasn't been precisely defined. Now, some of the time in math it's clear what we mean when we write some complicated expression; however, this isn't one of those times. We need to sit down and precisely define what this thing is.

When we try to do so, we'll find something surprising: it doesn't follow the usual laws of arithmetic! For example, consider the following "proof" that $1=2$: $${1\cdot{\infty\over\infty}}={\infty\over\infty}={2\infty\over \infty}=2\cdot{\infty\over\infty},\mbox{ so }1=2.$$ Since $1\not=2$, at least one of the steps there has to be nonsense. The obvious candidate is the claim that we can cancel $\infty\over\infty$. This suggests that ${\infty\over\infty}=0$; however, then we have $\infty\cdot {1\over\infty}\not=1$, which ruins one of the basic properties of division!

It gets worse: in both arguments, we need to perform infinitely many operations (either infinitely many sums, or infinitely many products), so we need to define how those work. At first glance it seems like limits can help us there, but again, we'll find that there's no good way to define the infinite operations we need which matches with our intuition.

This all points to the following fact:

In order to make sense of arithmetic involving infinity or infinite operations, we need to make some choices in how we precisely define the various operations on infinity; and no matter how we make these choices, some "obvious" properties of numbers will fail to hold.

Picture trying to fit a carpet into a room that's too small. You can maybe get lots of it to lie flat, but somewhere it's going to fold over, or stick up, or crumple.

The problem is that before you can even consider a proof of a statement, it has to be clear what that statement means. I have no idea what $\infty/\infty=0$ could possibly mean.

The original answer ended here.

Since not everyone was happy with that answer, let me elaborate. Since $\infty$ is not a number, I just do not know what $\infty/\infty$ should mean. Now you could respond that you are using formulas like $\infty+\infty=\infty$ and have been told that that one is true. Well, it is a useful short-hand, but what people really mean when they write this is the following:

If $(a_n)$ and $(b_n)$ are sequences of real numbers such that $\lim_{n\to\infty} a_n=+\infty$ and $\lim_{n\to\infty} b_n=+\infty$, then $\lim_{n\to\infty} a_n+b_n=+\infty$.

Now this is a perfectly fine statement, and it happens to be also true. Another perfectly fine statement (but a wrong one) would be:

If $(a_n)$ and $(b_n)$ are sequences of real numbers such that $\lim_{n\to\infty} a_n=+\infty$ and $\lim_{n\to\infty} b_n=+\infty$, then $\lim_{n\to\infty} a_n/b_n=0$.

Now maybe one could understand $\infty/\infty=0$ in this way, but you would have to say so. And now you also see that it would be hard to see how the “proof” in this question relates to this. Maybe one could make the following weaker statement:

If $(b_n)$ be a sequence of real numbers such that $\lim_{n\to\infty} b_n=+\infty$, then $\lim_{n\to\infty} n/b_n=0$.

One could then say that the “proof” in your question is to be understood as follows: \begin{multline}\lim_{n\to\infty} \frac{n}{b_n} =\lim_{n\to\infty}\underbrace{\left(\frac1{b_n}+\cdots+\frac1{b_n}\right)}_{\text{$n$ summands}} =\\=\lim_{n\to\infty}\underbrace{\left(\lim_{n\to\infty} \frac1{b_n}+\cdots+\lim_{n\to\infty}\frac1{b_n}\right)}_{\text{$n$ summands}} =\\=\lim_{n\to\infty}\underbrace{(0+\cdots+0)}_{\text{$n$ summands}}=\lim_{n\to\infty}0=0. \end{multline} Now here at least every expression has a defined meaning and we can ask: Where is this wrong? Well, it turns out that the second equality is wrong, and the reason why is again boring: There is just no reason that this equality should hold. It is not enough that it somehow looks nice, we would have to cite some theorem that we have proved earlier to justify this equality, and there is none. (Of course we can pinpoint the wrong equation by plugging in a counterexample. For example for $b_n$=n you can evaluate each expression and get $1=1=0=0=0=0$.)

Anyway, my point is that before we can ask why something is wrong it first has to make minimal sense. And the proof in your question is, as they say, “not even wrong”, because it is unclear what the statement is.