# Thread: interval on real number line

1. ## interval on real number line

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

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

My thinking is that $\frac{1}{n}$ 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 $[1, 5+\epsilon)$. If I drop the $\epsilon$, I will still get [0,5].

I am not too sure. I need help.

2. Yes, it's true. For any number larger than 5 (call it 'x'), you can always find a value of n such that x is not in the interval [1,5+1/n). Therefore, it becomes a closed interval [1,5].

Another way to say this is:

$\bigcap_{n=1}^{\infty} ~ \left[1,5+\frac{1}{n}\right) = [1,5]$

3. Originally Posted by drumist
Yes, it's true. For any number larger than 5 (call it 'x'), you can always find a value of n such that x is not in the interval [1,5+1/n). Therefore, it becomes a closed interval [1,5].

Another way to say this is:

$\bigcap_{n=1}^{\infty} ~ \left[1,5+\frac{1}{n}\right) = [1,5]$
Thank you for giving the logical explanation.