difference between nested interval theorem and cantor intersection theorem What is difference between the nested interval theorem and Cantor intersection theorem? In $\mathbb{R}$, are they the same? When are they different? 
Intuitively, I feel they are same for $\mathbb{R^n}$. But what happens in a general space?
Nested Interval Theorem
  Consider a family of closed and bounded real intervals, $I_1 = [a_1, b_1], I_2 = [a_2, b_2], \ldots$. 
  If $a_n \leq a_{n+1}$ and $b_{n+1} \leq b_n$ for all $n$ then there is an $x$ which is in every $I_n$, that is, $\displaystyle\bigcap_{n=1}^{\infty} I_n \neq \emptyset$.
  
  Cantor's Intersection Theorem
  Let S be a complete metric space. Consider a decreasing, nested sequence $\{C_k\}$ of nonempty closed and bounded sets in $S$. Then, $\displaystyle\bigcap_{k=1}^{\infty} C_k \neq \emptyset$.


It seems to me that Cantor's Intersection Theorem generalizes the Nested Interval Theorem to arbitrary complete spaces.

To see why the completeness hypothesis is necessary, consider $\{C_k\}={[0,\frac{1}{k}]}$. If $S=\mathbb{R}$, then $\displaystyle\bigcap_{k=1}^{\infty} C_k = \{0\}$. If $S=\mathbb{R}-\{0\}$, then $\displaystyle\bigcap_{k=1}^{\infty} C_k = \emptyset$.
Nested Interval Theorem
  Consider a family of closed and bounded real intervals, $I_1 = [a_1, b_1], I_2 = [a_2, b_2], \ldots$. 
  If $a_n \leq a_{n+1}$ and $b_{n+1} \leq b_n$ for all $n$ then there is an $x$ which is in every $I_n$, that is, $\displaystyle\bigcap_{n=1}^{\infty} I_n \neq \emptyset$.
  
  Cantor's Intersection Theorem
  Let S be a complete metric space. Consider a decreasing, nested sequence $\{C_k\}$ of nonempty closed and bounded sets in $S$. Then, $\displaystyle\bigcap_{k=1}^{\infty} C_k \neq \emptyset$.


It seems to me that Cantor's Intersection Theorem generalizes the Nested Interval Theorem to arbitrary complete spaces.

To see why the completeness hypothesis is necessary, consider $\{C_k\}={[0,\frac{1}{k}]}$. If $S=\mathbb{R}$, then $\displaystyle\bigcap_{k=1}^{\infty} C_k = \{0\}$. If $S=\mathbb{R}-\{0\}$, then $\displaystyle\bigcap_{k=1}^{\infty} C_k = \emptyset$.