Check if Cantor’s Theorem is applicable in the following examples. Also, find $\bigcap_{n\in\Bbb{N}} F_n$ in each case So we have to find whether Cantor's Theorem is applicable to the following
(i) $F_n = [n, \infty)$ (ii) $F_n = (0, \frac{1}{n})$ (iii) $F_n = [1 − \frac{1}{n} , 2 + \frac{1}{n}]$
Now I've tried doing these as

$F_n$ is non-empty and closed and $[n+1,\infty)\subset[n,\infty)$. But since $n\to\infty$, the length of $F_n$ does not tend to $0$. So not applicable
$F_n$ is an open interval so not applicable
$F_n$ is non-empty and closed. But as $n\to\infty$, $F_n\to [1,2]$ and length $[1,2]$ is not equal to $0$. So not applicable.

Is my method correct or am I doing something wrong??
Note that the exact statement of Cantor’s Intersection Theorem is the following:

The intersection of any decreasing sequence of non-empty nested compact sets is non-empty.

Thus, as you correctly stated, it is not applicable in cases a) and b). However, it is clearly applicable in case c), since that sequence satisfies all of the assumptions of Cantor’s Theorem.
Note that the exact statement of Cantor’s Intersection Theorem is the following:

The intersection of any decreasing sequence of non-empty nested compact sets is non-empty.

Thus, as you correctly stated, it is not applicable in cases a) and b). However, it is clearly applicable in case c), since that sequence satisfies all of the assumptions of Cantor’s Theorem.