Real Analysis

Math 310  MWF 10:30-11:25am
Fall 1996  127 Fowler

Ron Buckmire ∼ 402 Fowler ∼ 259-2536 ∼ ron@oxy.edu ∼ Mon 1:30-3:30pm Wed & Thu 1:30-2:30pm

#15 Wednesday (10/16/96)

SUMMARY Cauchy Sequences

READING pages 46-47

\underline{Warm-Up} (5-10 minutes) Recall the definition of a Cauchy Sequence

\underline{Example} Let’s show that the sequence s_{n} = 1/n is Cauchy.

\underline{Groupwork} Form three groups of two and in each group think of a convergent sequence. Use the formal definition of what a Cauchy sequence is to show that your convergent sequence is Cauchy.

When we say that a sequence is Cauchy we have made quite a strong statement about the sequence. [Think about the connections between statements such as “This sequence is Cauchy.” “This sequence is convergent.” “This sequence is bounded.”] \underline{Theorem} 10.9 All convergent sequences are Cauchy sequences. \underline{Theorem}

10.10 All Cauchy sequences are bounded. \underline{Theorem}

10.11 A sequence is convergent if and only if it is a Cauchy sequence.

Subsequences Given a sequence (s_{n}) a subsequence of this sequence is a sequence (t_{k}) where for each k there is a positive integer n_{k} such that t_{k} = s_{n_{k}}. (Note that n, k ∈ ℕ and and n_{1} < n_{2} < ⋯ < n_{k} < n_{k+1} < ⋯) Often the subset of ℕ consisting of {n_{1}, n_{2}, n_{3}, …} is computed using a “selection function” denoted by σ(k) = n_{k}. \underline{Example} Let (s_{n}) be the sequence defined by s_{n} = n^{2}(−1)^{n} The positive elements of this sequence make up a subsequence.

• Write down the first 8 terms of (s_{n})

• Write down the first 8 terms of (s_{n_{k}})

• Write down an expression for n_{k} in terms of k (i.e. σ(k))

• Write down a definition of the k-th term of (s_{n_{k}})

How many subsequences does a sequence have? Think of one other subsequence of the above sequence.

\underline{Theorem} 11.2 If A sequence (s_{n}) converges, then every subsequence (s_{n_{k}}) converges to the same limit

Proof:

Homework Assignment (Due Monday October 21) §10, page 47-48: 10.1, 10.6 10.10 §11, page 55: 11.2