4. Limits of Sequences

Calculus 1



Weiqi Zhou

Zeno's Paradox

Limit of a Sequence

  • given a real sequence and a

  • if for any , there exists some , so that holds for all

  • then is said to be convergent, with limit

Example

  • , verify by definition that

  • given an arbitrary ，let be a number satisfying

  • if , then

Exercises

show that

Example：0.9999....=1.00000....

The representation of a number in base is not unique

Adjusting finitely many terms of a sequence does not change its convergence or limit

Uniqueness

If converges, then it has a unique limit

Proof of Uniqueness

  • suppose that both are limits of

  • for any , and sufficiently large , we have ，



  • since is arbitrary, this implies that

Boundedness

if is convergent, then there exists ，so that holds for all

Proof of Boundedness

  • suppose that is the limit of

  • take ，then there exists ，so that holds for all

  • consequently

  • it then suffices to take

Convergence of Subsequences

if is convergent，
then so are all its subsequences，
and they share the same limit as

Exercises

  • is convergent? why?

  • is bounded，，show that

  • is arbitrary，，is convergent? why?

  • if a subsequence of converges, does it imply convergence of ? why?

Criterion: Squeezing

if satisfy

then

Proof

  • given ，there exist ，so that and for all

  • take ，then for all any we have


  • it follows that

Exercise

Monotonicity + Boundedness

  • the following two types of sequences are convergent:

  • a monotonically non-decreasing upper bounded sequence

  • a monotonically non-increasing lower bounded sequence

Exercises

  • suppose ,

  • prove by induction that

  • prove that is monotonically decreasing

  • compute , what might be the limit of this sequence?

Summary

  • Definition

  • Properties

  • Criteria