MAT425 Real Analysis L14

Prof. Thistleton

The Squeeze Theorem Recall from Calculus I that

$$\lim_{x \rightarrow 0 } \frac{sin(x)}{x} = 1$$

and that the proof of this theorem requires a little trigonometry and the squeeze theorem for functions. What is the squeeze theorem for functions?

How would you state a similar result for sequences?

We present a proof of the squeeze theorem for sequences.

Theorem 210

For the following, suppose that and that .

• If for all n then .

  
  
  
  
  
  
  
  
  
  
  
  
  
  

• If for all n, then .

  
  

• If L_x = L_y = L and if for all n, then .

  
  
  
  
  
  
  
  
  
  
  
  
  
  

Example

Sequences behave pretty much as we would like them to with respect to basic arithmetic. We state and prove results below concerning sequences and addition and multiplication.

Theorem 219

For the following, suppose that and that .

• lim(x_n + y_n ) = L_x + L_y.

  
  

• lim(c x_n) = c L_x.
  
  
  
  
  
  
  
  
  
  
  
  
  
• lim(x_n y_n ) = L_x L_y.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
• lim(x_n / y_n ) = L_x / L_y if and .

We now combine what we have learned about sequences with our ideas about topology. Recall that a point x is a cluster point of the set A if, for every , the intersection of the -neighborhood around x and the set A is nonempty. That is, .We have the following definition.

Definition 230

A point x is said to be a cluster point of the sequence (x_n) if it is a cluster point of the range of (x_n).

Lemma 235

Suppose that x_n is a convergent sequence with limit L. Then x_n has at most one cluster point, L.

Examples

• Is it true that every convergent sequence has a cluster point?

  
  
  
  
  
  
  
  

• Is it true that every sequence with a cluster point has a limit?

  
  
  
  
  
  
  
  

• Is it true that a sequence can have at most one cluster point?

The preceeding discussion leads us to the following theorem.

Theorem 242

Suppose that (x_n) is a convergent sequence with limit L. Then one of the following is true:

• (x_n) is eventually equal to L.
• L is the only cluster point of (x_n).

We are almost in a position to characterize closed sets in in terms of convergent sequences. We need one more result.

Lemma 249

The point c is a cluster point of the set S if and only if there is a sequence of elements of S, all different from c, and converging to c.

Theorem 255

A subset S of is closed if and only if whenever (x_n) is a convergent sequence whose terms are all in S.

Before proving the result state this result in plain english.