Another marble and urn problem, this time to $\infty$

$$\Pr(\text{1 is picked up in some time}){=1-\Pr(\text{1 is not picked up in any time})\\=1-{1\over 2}\times{2\over 3}\times{3\over 4}\times\cdots\\=1-\lim_{n\to \infty}{1\over n+1}\\=1}$$

Let $E_n$ be the event "you haven't drawn the $1$ in the first $n$ trials."

Then $$P(E_n)=\prod_{i=2}^n \frac {i-1}i=\frac 1n $$

Where, for the last equality we used the fact that the product telescopes.

It is clear, therefore, that $$\lim_{n\to \infty} P(E_n)=0$$ and we are done.