Squeeze Theorem/Sequences

Theorem

There are two versions of this result:

• one for sequences in the set of complex numbers $\C$, and more generally for sequences in a metric space.
• one for sequences in the set of real numbers $\R$, and more generally in linearly ordered spaces (which is stronger).
  

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  Sequences of Real Numbers

  Let $\sequence {x_n}$, $\sequence {y_n}$ and $\sequence {z_n}$ be sequences in $\R$.

  Let $\sequence {y_n}$ and $\sequence {z_n}$ both be convergent to the following limit:

  $\ds \lim_{n \mathop \to \infty} y_n = l, \lim_{n \mathop \to \infty} z_n = l$

  Suppose that:

  $\forall n \in \N: y_n \le x_n \le z_n$

  
  Then:

  $x_n \to l$ as $n \to \infty$

  that is:

  $\ds \lim_{n \mathop \to \infty} x_n = l$

  
  Thus, if $\sequence {x_n}$ is always between two other sequences that both converge to the same limit, $\sequence {x_n} $ is said to be sandwiched or squeezed between those two sequences and itself must therefore converge to that same limit.

  
  

  Sequences of Complex Numbers

  Let $\sequence {a_n}$ be a null sequence in $\R$, that is:

  $a_n \to 0$ as $n \to \infty$

  Let $\sequence {z_n}$ be a sequence in $\C$.

  
  Suppose $\sequence {a_n}$ dominates $\sequence {z_n}$.

  That is:

  $\forall n \in \N: \cmod {z_n} \le a_n$

  
  Then $\sequence {z_n}$ is a null sequence.

  
  

  Sequences in a Linearly Ordered Space

  Let $\struct {S, \le, \tau}$ be a linearly ordered space.

  Let $\sequence {x_n}$, $\sequence {y_n}$, and $\sequence {z_n}$ be sequences in $S$.

  Let $p \in S$.

  Let $\sequence {x_n}$ and $\sequence {z_n}$ both converge to $p$.

  Let $\forall n \in \N: x_n \le y_n \le z_n$.

  
  Then $\sequence {y_n}$ converges to $p$.

  
  

  Sequences in a Metric Space

  Let $M = \struct {S, d}$ be a metric space or pseudometric space.

  Let $p \in S$.

  Let $\sequence {r_n}$ be a null sequence in $\R$.

  Let $\sequence {x_n}$ be a sequence in $S$ such that:

  $\forall n \in \N: \map d {p, x_n} \le r_n$.

  
  Then $\sequence {x_n}$ converges to $p$.

  
  

  Also known as

  This result is also known, in the UK in particular, as the sandwich theorem.