Real Analysis (all - semester 1)

How is the set of rational numbers Q defined?

Q = {p/q : p, q ∈ Z, q ≠ 0};;

What is the problem with Q from the point of view of analysis?

It has "gaps": e.g. the rational sequence 1, 1.4, 1.41, 1.414, … (truncations of √2) has a limit that isn't rational;;

How do you prove √2 is not rational?

1. Suppose √2 = p/q with p, q ∈ N coprime

2. Square: p² = 2q², so p is even, p = 2k

3. Then q² = 2k², so q is even too

4. So 2 is a common factor of p and q: contradiction;;

When is a set X ⊆ R bounded above?

If there exists M ∈ R with x ≤ M for all x ∈ X;;

What is an upper bound for a non-empty set X bounded above?

Any M ∈ R with x ≤ M for all x ∈ X;;

What is the supremum of a non-empty set X bounded above?

The least upper bound sup X:

(a) sup X is an upper bound for X

(b) sup X ≤ M for every upper bound M of X

Equivalently, sup X = min{M : M is an upper bound for X};;

What is sup X if X is not bounded above, and what is sup ∅?

sup X = ∞ if X isn't bounded above; sup ∅ = −∞;;

When is X ⊆ R bounded below, and what is inf X?

Bounded below: there is m ∈ R with x ≥ m for all x ∈ X. inf X = the greatest lower bound. If X isn't bounded below, inf X = −∞; inf ∅ = ∞;;

When is a set X ⊆ R bounded?

When it's bounded above and below: there exist m, M with m ≤ x ≤ M for all x ∈ X;;

How are sup and max related?

If X has a largest element, sup X = max X; sup is a substitute for max when there's no largest element (similarly inf X = min X if X has a smallest element);;

What are sup and inf of [0, 1]?

sup = 1, inf = 0 (the max and min);;

How do you prove sup (0, 1) = 1?

1. 1 is an upper bound, so sup X ≤ 1

2. Suppose sup X = a < 1

3. Then (a + 1)/2 ∈ (0, 1) and (a + 1)/2 > a, so a isn't an upper bound: contradiction;;

What are sup and inf of X = {(n + 1)/n : n ∈ N}, and how do you prove the inf?

sup X = max X = 2 (at n = 1, since 1 + 1/n is decreasing). inf X = 1:

1. 1 is a lower bound, so inf X ≥ 1

2. Suppose inf X = a > 1; then 1 + 1/n ≥ a for all n

3. Rearranging: n ≤ 1/(a − 1) for all n, contradicting that N isn't bounded above;;

What three types of axioms make R a complete ordered field?

Algebraic axioms (+ and · make R a field), ordering axioms (< makes it an ordered field), and the Completeness Axiom;;

What is the Completeness Axiom of R (Least Upper Bound Property)?

If X ⊆ R is non-empty and bounded above, then sup X exists in R;;

Which axiom distinguishes R from Q?

The Completeness Axiom: Q is also an ordered field but lacks it;;

What is the equivalent version of the Completeness Axiom using lower bounds?

If X ⊆ R is non-empty and bounded below, then inf X exists in R;;

What set shows Q doesn't have the least upper bound property?

{x ∈ Q : x² < 2} has no least upper bound in Q (it does in R);;

How do you prove there is a real number x with x² = 2 (sketch)?

1. Let X = {x ∈ R : x² < 2}: non-empty (0 ∈ X) and bounded above by 2

2. By the Completeness Axiom, a = sup X exists

3. Rule out a² < 2: find small ε > 0 with (a + ε)² < 2, contradicting a being an upper bound

4. Rule out a² > 2: find ε > 0 with (a − ε)² ≥ 2, contradicting a being the least upper bound;;

What are the drawbacks of defining R as infinite decimals?

Decimal expansions aren't unique (37.5399… = 37.54000…), and adding infinite decimals can involve carrying indefinitely;;

How is the modulus |x| defined?

|x| = x if x ≥ 0, and −x if x < 0;;

What are the basic properties of |x|?

|x| ≥ 0, with |x| = 0 iff x = 0; √(x²) = |x|; −|x| ≤ x ≤ |x|; for ε > 0, |x| < ε iff −ε < x < ε; |xy| = |x||y|;;

What is the distance between x and y in R?

|x − y|;;

What is the Triangle Inequality?

|a + b| ≤ |a| + |b| for all a, b ∈ R;;

How do you prove the Triangle Inequality?

1. ab ≤ |ab| = |a||b|

2. So a² + 2ab + b² ≤ |a|² + 2|a||b| + |b|²

3. i.e. |a + b|² ≤ (|a| + |b|)²

4. Take positive square roots;;

What is the formal definition of a real sequence?

A function from N to R, n ↦ aₙ, written (aₙ) or (aₙ)ₙ₌₁^∞;;

When would you write (aₙ)ₙ₌₀^∞ or (aₙ)ₙ₌₅^∞?

When the sequence starts at a different index (e.g. a₀ or a₅);;

Is listing the first few terms enough to describe a sequence?

No, every term must be described, e.g. by a formula for the nth term or a recursive formula;;

What does it mean for a sequence (aₙ) to tend to infinity?

Given any A > 0 there exists N ∈ N such that aₙ > A for all n > N (∀A > 0 ∃N ∈ N s.t. aₙ > A ∀n > N);;

In the definition of aₙ → ∞, what does N depend on?

On A: the larger A, the further along you generally need to go;;

Do you need the smallest N when proving a limit from the definition?

No, any N that works is enough;;

How do you prove √n → ∞?

Let A > 0. Choose N ∈ N with N ≥ A². Then n > N ⇒ n > A² ⇒ √n > A;;

How do you prove (n² + n)/(n + 5) → ∞?

1. Bound below: (n² + n)/(n + 5) ≥ n²/(n + 5) ≥ n²/(n + 5n) = n/6

2. Let A > 0; choose N ≥ 6A

3. n > N ⇒ n/6 > A ⇒ aₙ > A;;

What trick helps find simple inequalities for fractions a/b with a, b > 0?

a/b shrinks if a shrinks or b grows, and grows if a grows or b shrinks;;

If aₙ ≥ bₙ for all n ≥ N₀ and bₙ → ∞, what can you conclude?

aₙ → ∞;;

How do you prove that aₙ ≥ bₙ (n ≥ N₀) and bₙ → ∞ imply aₙ → ∞?

Let A > 0. Get N′ with bₙ > A for n > N′. Set N = max{N′, N₀}; then aₙ ≥ bₙ > A for all n > N;;

How do you prove 2√n + 5(−1)ⁿ → ∞?

5(−1)ⁿ ≥ −5 ≥ −√n for n ≥ 25, so aₙ ≥ √n for n ≥ 25; since √n → ∞, the comparison result with N₀ = 25 gives aₙ → ∞;;

What does it mean for a sequence to tend to minus infinity?

Given any A > 0 there exists N ∈ N with aₙ < −A for all n > N; equivalently −aₙ → ∞;;

What does it mean for (aₙ) to converge to ℓ ∈ R?

Given any ε > 0 there exists N ∈ N such that |aₙ − ℓ| < ε for all n > N (∀ε > 0 ∃N ∈ N s.t. |aₙ − ℓ| < ε ∀n > N);;

Can ±∞ be the limit of a convergent sequence?

No, ℓ must be in R, and ±∞ are not real numbers;;

What is |aₙ − ℓ| < ε equivalent to?

ℓ − ε < aₙ < ℓ + ε;;

In the definition of convergence, what does N depend on?

On ε: the smaller the allowed error ε, the further along you generally need to go;;

How do you prove (n + 1)/n → 1?

1. |aₙ − 1| = 1/n

2. Let ε > 0; choose N ∈ N with N > 1/ε

3. n > N ⇒ |aₙ − 1| = 1/n < 1/N < ε;;

How do you prove (n² + n + 1)/(2n² + 1) → 1/2 from the definition?

1. |aₙ − 1/2| = (2n + 1)/(2(2n² + 1)) < (2n + 1)/(4n²) ≤ 3n/(4n²) = 3/(4n)

2. Let ε > 0; choose N > 3/(4ε)

3. n > N ⇒ |aₙ − 1/2| < 3/(4n) < 3/(4N) < ε;;

How do you structure an ε-N proof that aₙ → ℓ?

1. Simplify |aₙ − ℓ| and bound it by something simple that tends to 0 (e.g. c/n)

2. "Let ε > 0 be arbitrary"

3. Choose N from the simple bound (e.g. N > c/ε)

4. Show n > N ⇒ |aₙ − ℓ| < ε

5. Conclude since ε was arbitrary;;

Can a convergent sequence have two different limits?

No, limits are unique;;

How do you prove a convergent sequence has a unique limit?

1. Suppose aₙ → ℓ and aₙ → m with ℓ ≠ m; let ε = |ℓ − m|/4

2. Get N₁, N₂ with |aₙ − ℓ| < ε and |aₙ − m| < ε

3. For n > max{N₁, N₂}: |ℓ − m| ≤ |aₙ − ℓ| + |aₙ − m| < 2ε = |ℓ − m|/2

4. Contradiction;;

When is a sequence bounded above, bounded below, bounded?

Bounded above: aₙ ≤ M for all n. Bounded below: aₙ ≥ M for all n. Bounded: M₁ ≤ aₙ ≤ M₂ for all n, equivalently |aₙ| ≤ M for all n ∈ N;;

Does every bounded sequence converge?

No, e.g. aₙ = (−1)ⁿ is bounded (|aₙ| ≤ 1) but doesn't converge;;

Is every convergent sequence bounded?

Yes;;

How do you prove every convergent sequence is bounded?

1. Use the definition with ε = 1: |aₙ − ℓ| < 1 for n > N

2. Triangle inequality: |aₙ| ≤ |aₙ − ℓ| + |ℓ| < 1 + |ℓ| for n > N

3. Take M = max{|a₁|, …, |a_N|, 1 + |ℓ|}, so |aₙ| ≤ M for all n;;

What is the Algebra of Limits?

If aₙ → ℓ and bₙ → m, then:

(i) aₙ + bₙ → ℓ + m

(ii) λaₙ → λℓ for λ ∈ R

(iii) aₙbₙ → ℓm

(iv) aₙ/bₙ → ℓ/m if m ≠ 0;;

How do you prove aₙ + bₙ → ℓ + m?

1. Let ε > 0; get N₁, N₂ with |aₙ − ℓ| < ε/2 and |bₙ − m| < ε/2

2. N = max{N₁, N₂}

3. |aₙ + bₙ − (ℓ + m)| ≤ |aₙ − ℓ| + |bₙ − m| < ε for n > N;;

How do you prove aₙbₙ → ℓm?

1. (bₙ) converges so is bounded: |bₙ| ≤ M

2. Let ε > 0; get |aₙ − ℓ| < ε/(2M) and |bₙ − m| < ε/(2(1 + |ℓ|)) for n > N

3. |aₙbₙ − ℓm| = |(aₙ − ℓ)bₙ + (bₙ − m)ℓ| ≤ |aₙ − ℓ|M + |bₙ − m||ℓ| < ε;;

In the product rule proof, why use ε/(2(1 + |ℓ|)) rather than ε/(2|ℓ|)?

ℓ could be 0, and you can't divide by 0;;

How do you use the Algebra of Limits to show (n² + n + 1)/(2n² + 1) → 1/2?

Divide top and bottom by n²: (1 + 1/n + 1/n²)/(2 + 1/n²). Since 1/n → 0, AOL for products, sums and quotients gives (1 + 0 + 0)/(2 + 0) = 1/2;;

If aₙ → ℓ, bₙ → m and aₙ ≤ bₙ for all n ≥ N₀, what can you conclude?

ℓ ≤ m;;

If aₙ ≤ M for all n and aₙ → ℓ, what can you conclude?

ℓ ≤ M (and similarly a lower bound M gives ℓ ≥ M): the limit satisfies any bound the sequence satisfies;;

If aₙ < bₙ for all n, must lim aₙ < lim bₙ?

No, e.g. −1/n < 1/n but both limits are 0; strict inequalities aren't preserved;;

If aₙ → ℓ > 0, what can you say about aₙ for large n?

There's N with aₙ > 0 for all n > N (similarly ℓ < 0 gives aₙ < 0 eventually);;