Find the number of members in a sequence $X_n=\frac{2n-1}{4n+5}$, outside the interval $[\frac{1}{2}-\frac{1}{1000},\frac{1}{2}+\frac{1}{1000}]$ Find the number of members in a sequence $X_n=\frac{2n-1}{4n+5}$ outside the interval $ [\frac{1}{2}-\frac{1}{1000} , \frac{1}{2} + \frac{1}{1000}]$.
Step 1:
I remember these three definitions.
$ \lim X_{n\to \infty} = \ell $
$ |X_n -\ell|\lt\varepsilon$
$(\ell+\varepsilon), (\ell - \varepsilon) $
Step 2:
I substitute the desired values.
$ \lim X_{n\to \infty} = \frac{1}{2} $
$ |\frac{2n-1}{4n+5} -\frac{1}{2}|\lt\frac{1}{1000} $
Step 3:
I calculate
$|\frac{2n-1}{4n+5} -\frac{1}{2}|\lt\frac{1}{1000} \Rightarrow$
$ |\frac{2(2n-1)}{2(4n+5)} -\frac{1(4n+5)}{2(4n+5)}|\lt\frac{1}{1000} \Rightarrow$
$ |\frac{4n-2-4n-5}{8n+10}|\lt\frac{1}{1000} \Rightarrow$
$ |\frac{-7}{8n+10}|\lt\frac{1}{1000} \Rightarrow$
$ \frac{7}{8n+10}\lt \frac{1}{1000} \Rightarrow$
$ 6990\lt 8n $
Step 4:
I have no idea what to do next.
$n$ by definition, the number of the element with which the elements that lie inside the interval begin. I am asked to find how many elements are outside the interval.
That is, I need something like $n-1$. Since it is out of range.
That is, if I know that $8$ elements will lie inside the interval, then $7$ will lie outside the interval.
But before that, I need to find the extreme $n$, which will determine the members of the sequence that lie within the interval and also satisfy the inequality. After that, find the desired $n - 1$, from which the terms will lie outside the interval.
Step 5:
$ 6990\lt 8n \Rightarrow 873.75\lt n $
$ 873.75 \approx 874 $
$ 873.75 \lt n $
$ 873.75 \lt 874 $
$ n-1 \Rightarrow 874 - 1 = 873 $
Answer: $873$
The main question: this is that number $873$ that is outside the range? Did I do everything right or was I wrong? Please, say Me.

You have found
$$|\frac{2n-1}{4n+5} -\frac{1}{2}|<\frac{1}{1000} \iff n > 873,75.$$
Since $ n \in \mathbb N$, this gives
$$|\frac{2n-1}{4n+5} -\frac{1}{2}|<\frac{1}{1000} \iff n  \ge  874.$$
Hence
$$|\frac{2n-1}{4n+5} -\frac{1}{2}| \ge \frac{1}{1000} \iff n  \le  873.$$
