Understanding Infinite Sequences: Difference of 1 & n/(n+1)

[find_the_fun]

My textbook reads :

The graph of \(\displaystyle a_n=\frac{n}{n+1}\) are approaching 1 as n becomes large . In fact the difference
\(\displaystyle 1-\frac{n}{n+1}=\frac{1}{n+1}\) can be made as small as we like by taking n sufficently large. We indicate this by writing \(\displaystyle \lim_{n \to \infty} \frac{n}{n+1}=1\)

I don't understand where they pull \(\displaystyle \frac{1}{n+1}\) from and what difference they refer to?

 

[Siron]

They refer to the difference
$$1-\frac{n}{n+1}$$
If $n \to \infty$ we have $\frac{1}{n+1} \to 0$ and thus $1-\frac{n}{n+1} \to 0$. In other words, the difference can be made as small as you like by taking $n$ sufficently large.