Test #3 Elementary Analysis

Summation Notation

∑ak​=am​+am+1​+⋯+an​

It tells you to add terms from index k=m to k=n

Subsequence

A subsequence of (s_n) is a sequence (s_nk) where (nk) is a strictly increasing sequence of natural numbers:
n₁<n₂<n₃<⋯

Give equivalent ways to define a subsequence

• Determined by a strictly increasing sequence (n_k)

• Any infinite subset of ℕ (in increasing order)

Functional Notation of a Subsequence

What does a subsequence do intuitively?

It picks terms from the original sequence without changing order.

When does a subsequence converge to t? Subsequence Limit Theorem (Part i)

A subsequence converges to t iff for every ε>0, the set
{n: ∣s_n−t∣ < ε} is infinite

What happens if a sequence is unbounded above? Subsequence Limit Theorem (Part ii)

It has a subsequence that diverges to +∞

What happens if a sequence is unbounded below? Subsequence Limit Theorem (Part iii)

It has a subsequence that diverges to -∞

What extra property can subsequences have in Theorem 0.1? Subsequence Limit Theorem (Part iv)

They can be chosen to be monotonic (increasing or decreasing)

If s_n→s, what happens to subsequences? Subsequence of Convergent Sequence

Every subsequence s_nk→s

Why does convergence pass to subsequences?

Because n_k≥k so subsequences eventually follow the same tail behavior.

Monotone Subsequence Theorem

Every sequence has a monotonic subsequence (either increasing or decreasing).

What is a “peak”?

An index n where s_n is greater than all terms after it.

How does the proof split? (peaks)

• Infinitely many peaks → decreasing subsequence

• Finitely many peaks → increasing subsequence

Subsequential Limit

Any value in R ∪ {±∞} that is the limit of some subsequence.

How are supS and inf⁡S related?

supS=limsups_n

infS=liminfs_n

When does a sequence converge using S?

The sequence converges iff S has exactly one element

limsup/liminf Subsequences

There exists:

• A subsequence → lim sup⁡s_n

• A subsequence → lim inf⁡s_n
  Both can be monotonic