$0.00\dots 1=0$ claim

Question

I was reading this webpage about a claim to do with $0 \times \infty $. In it he states in red that $0.00\dots 1=0$ for the same reasons that $0.9999\dots= 1$. I fail to see his logic and his supposed proof that he provides (also in red) is just a Wikipedia article proving $0.9999\dots = 1$. Can anyone explain whats going on? Is this guy talking rubbish or is there some truth to his argument?

Answer 1

The only truth is that if you interpret his "number" the only way possible,

$$\lim_{n \to \infty} \frac1{10^n}$$

then indeed this limit is $0$. But his notation is completely whack in my opinion, because you can't write a bunch of zeros and then literally write a $1$ at position "infinity"