summation in limits

summation in limits

I was reading a solution to a problem given in my book the problem was
$$\lim_{n \to \infty}\left( \frac{1}{n + 1}+ \frac{1}{n + 2} + \frac{1}{n
+ 3 }+........+\frac{1}{6n}\right)$$ now it was then converted to $$
\lim_{n \to \infty}\sum_{r=1}^{5n}\left(\frac{1}{n+r} \right) =\lim_{n \to
\infty}\frac{1}{n} \sum_{r=1}^{5n}\left(\frac{1}{1+r/n} \right)$$ i
somewhat understand the conversion but what i don't get was that why
summation is done till $ 5n$ not $6n$ and finally it was written
$\int_{0}^{5}\frac{1}{1+x}dx$ i still somewhat get it buy why limits are
from $0$ to $5$ while in summation it's from $1$ to $5n$ please don't give
me external links and please explain me here