Suppose that I have an half open interval, $\displaystyle [1, 5+\frac{1}{n})$.

Is it correct to say that the half open interval will become a close interval, $\displaystyle [0,5]$ as n approaches $\displaystyle \infty$ ?

My thinking is that $\displaystyle \frac{1}{n}$ to the right of the number 5 changes, but it could never affect the number 5. In essence, at the end the interval will be $\displaystyle [1, 5+\epsilon)$. If I drop the $\displaystyle \epsilon$, I will still get [0,5].

I am not too sure. I need help.