I am currently learning about fractions, and there is something that I am finding it hard to make sense of. When a fraction it added to the right of the decimal point, the number becomes slightly bigger right? So for example if I add enough fractions to the 0.99999999, will this number eventually become equal to 1?
So will this number eventually become equal to 1 (if I added enough 9): 0.9999999999999999999999999999999999999999999999999999999999999999999999...9
Yes and no. When you are adding more $9$'s to the end of the number, you are adding $0.9$, $0.09$, or $9(0.1)^n$, where $n$ is the $n^{th}$ digit. So if we look at it with a geometric series, we find that adding $n \,9's=\sum_{k=1}^{n} 9\frac{1}{10}^k=\sum_{k= 0}^{n-1} \frac{9}{10} \left(\frac{1}{10}\right)^{k}= \dfrac{9}{10}\dfrac{1-\left(\frac{1}{10}\right)^{n-1}}{1-\frac{1}{10}}$ Let's see if this can equal $1$ for any $n$:
$$1=1-\frac{1}{10}^{n-1}\implies 0=-\frac{1}{10}^{n-1}$$This is never true because an exponential is always positive. So no matter how many $9$'s you add, you will never reach $1$. However, if you add infinite $9$'s, we can take the limit as $n\to \infty$ and see that infinite 9's does equal 1.
Here is a kind of proof with handmade animation, Hope it will be useful.
Another one is here too
