Real Analysis Cheat Sheet PDF

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Real Analysis 1.6.3 algebra of limits

definitions sequence: a function N → R value set: {an |n ∈ N} subsequence: (ank )n∈N with n1 < n2 < n3 < · · · converges to r: ∀ǫ > 0, ∃N ∈ N, ∀n ≥ N, |an − r| < ǫ divergent: not convergent upper bound of S: r ∈ R with S ≤ r lower bound of S: r ∈ R with r ≤ S S is bounded from above: S has an upper bound S is bounded from below: S has a lower bound bounded: bounded from above and from below

sum rule: lim (an + bn ) = lim (an ) + lim (bn ) n→∞

n→∞

n→∞

multiplication rule: lim (an · bn ) = lim (an ) · lim (bn ) division rule:

n→∞ n→∞ lim n→∞ (an ) lim ( abn ) = lim n→∞ (bn ) n→∞ n

n→∞

modulus rule: lim |an | = | lim an | n→∞ q n→∞ √ root rule: lim p an = p lim an , for p ∈ N n→∞

n→∞

propositions and friends

supremum: least upper bound, sup(S) infimum: greatest lower bound, inf(S) = -sup(-S)

1.2.7: every subsequence of a convergent sequence (an )n∈N converges to limn→∞ an 1.4.2: every convergent sequence is bounded 1.4.3: completeness axiom of R: every nonempty subset S of R which has an upper bound, has a least upper bound.

increasing: a1 ≤ a2 ≤ a3 ≤ · · · decreasing: a1 ≥ a2 ≥ a3 ≥ · · · strictly increasing: a1 < a2 < a3 < · · · strictly decreasing: a1 > a2 > a3 > · · · monotone: increasing or decreasing strictly monotone: strictly increasing or decreasing

properties of null sequences null sequence: sequence that converges to 0

Let (an )n∈N be a null sequence. (i) if c ∈ R then (c · an )n∈N (ii) if (bn )n∈N is null, then (an + bn )n∈N is null (iii) if |bn | ≤ |an | FABFM n, then (bn )n∈N is null (iv) if (bn )n∈N is bounded, then (an · bn )n∈N is null (v) if an ≥ 0 for all n, p ∈ R, then (apn )n∈N is null

1.2.5 uniqueness of limits if (an )n∈N → r and (an )n∈N → s then r = s

(an )n∈N → r ⇐⇒ (an − r)n∈N is null (an )n∈N is null ⇐⇒ (|an |)n∈N is null

1.2.6 finite modification rule if (an )n∈N → r and (bn )n∈N (an )n∈N for all but finitely many n then (bn )n∈N → r

=

1.4.6 monotone convergence theorem every monotone and bounded sequence has a limit

1.6.1 sandwich rule for sequences if (an )n∈N , (cn )n∈N → r and an ≤ bn ≤ cn for all n ∈ N then (bn )n∈N → r

1.6.2 compatibility of limits and order (i) if an ≤ bn FABFM n then limn→∞ an ≤ limn→∞ bn (ii) if limn→∞ an < limn→∞ bn then FABFM n an < bn

standard list of null sequences (i) limn→∞ n1p = 0 for every p ∈ R, p > 0 (ii) limn→∞ c1n = 0 for every c ∈ R, |c| > 1 p (iii) limn→∞ ncn = 0 for every p, c ∈ R, |c| > 1 n (iv) limn→∞ cn! = 0 for all c ∈ R

definitions

2.2.9 ratio test

n-th partial sum: sn = a1 + · · · + an P∞ the series n=1 converges: (sn )n∈N converges P∞ n=1 an = limn→∞ sn P∞ absolutely convergent: n=1 |an | is convergent

2.1.2 geometric series ∞ X

xk =

k=0

if (an )n∈N if a sequence with an 6= 0 for all n ∈ N such a |)n∈N is convergent. that (| n+1 an lim |

n→∞

∞ X an+1 an is absolutely convergent |1⇒ an n=1

1 for |x| < 1 1−x propositions and friends

2.1.4 (2.1.7 alternating) harmonic series ∞ X 1 is divergent k k=0

(alternating):

∞ X (−1)k+1 k=0

k

P∞ 2.1.3: if n=1 an converges, then (an )n∈N is null 2.2.3: every absolutely convergent series is convergent 2.2.4: 2.2.1 also holds for absolutely convergent P∞ convergent series and 2.2.6: if n=1 an is an absolutely P∞ (bn )n∈N is bounded, then n=1 (an · bn ) is absolutely convergent

is convergent repertoire

2.1.5 convergence of series with +ve terms if (an )n∈N is a sequence with an ≥ 0 for all n ∈ N then ∞ X

n=0

an is convergent ⇐⇒ (sn )n∈N is bounded

2.2.1 algebra of series P∞ P∞ (i) for every c ∈ R, n=0 P∞(c · an ) =Pc∞· n=0 an P∞ (ii) n=0 (an + bn ) = n=0 an + n=0 bn

2.2.5 comparison test P∞

if n=1 an is convergent and |bn | ≤ |an | for all Pabsolutely ∞ n ∈ N then n=1 bn is absolutely convergent

2.2.7 limit comparison test if an , bn > 0 for all n ∈ N and ( abnn )n∈N is convergent with limn→∞ abn 6= 0 then n

∞ X

n=1

an is convergent ⇐⇒

∞ X

n=1

bn is convergent

∞ X

1 is convergent for k > 1 k n n=0 ∞ n X c n=0

n!

is convergent for every c ∈ R

definitions

the boundedness theorem

continuous at x0 : S ∈ R, f : S → R, x0 ∈ S, then ∀ǫ > 0 ∃δ > 0 ∀x ∈ S(|x − x0 | < δ ⇒ |f (x) − f (x0 )| < ǫ) f tends to r from above: S ∈ R, f : S → R, a ∈ S with (a, a + h) ∈ S for some h > 0. Let r ∈ R then ∀ǫ > 0 ∃δ > 0 ∀x ∈ S (a < x < a + δ ⇒ |f (x) − r| < ǫ) f tends to r from below: S ∈ R, f : S → R, a ∈ S with (a − h, a) ∈ S for some h > 0. Let r ∈ R then ∀ǫ > 0 ∃δ > 0 ∀x ∈ S (a − δ < x < a ⇒ |f (x) − r| < ǫ)

if f : [a, b] → R is continuous, then f is bounded and f attains a global maximum and global minimum

deleted neighbourhood of a: (a − h, a) ∪ (a, a + h) limx→a f (x):= limxրa f (x) = limxցa f (x) if equal limx→∞ f (x) = r: f : S → R, (h, +∞) ∈ S. Then ∀ǫ > 0 ∃d ∈ R ∀x ∈ S (x > d ⇒ |f (x) − r| < ǫ) limxցaf (x) = ∞: f : S → R, (a, a + h) ∈ S for some a, h ∈ R. Then ∀A > 0 ∃δ ∈ R ∀x ∈ S (a < x < a + δ ⇒ f (x) > A)

characterisation of continuity via sequences let S ∈ R, x0 ∈ S, the following are equivalent: (i) f is continuous at x0 (ii) for every sequence (an )n∈N in S, if (an )n∈N → x0 then f ((an ))n∈N → f (x0 ) (iii) for every monotone sequence (an )n∈N in S, if (an )n∈N → x0 then f ((an ))n∈N → f (x0 )

algebra of limits for continuity at a point let S ∈ R, x0 ∈ S, f, g : S → R which are continuous at x0 then (i) f + g is continuous at x0 (ii) f · g is continuous at x0 (iii) if g(x) 6= 0, ∀x ∈ S, then fg is continuous at x0

composite rule for continuous functions let S, T ⊆ R, f : S → R, g : T → R with f (S) ⊆ T . if f is continuous at x0 and g is continuous at f (x0 ) then g ◦ f : S → R is continuous at x0

intermediate value theorem if f : [a, b] → R is continuous, then every number between f (a) and f (b) is attained by f . ∀r between f (a) and f (b), ∃c ∈ [a, b] with f (c) = r

algebra of limits of functions let S ⊆ R, f, g : S → R. let a ∈ R such that (a, a + h) ∈ S for some h > 0. if limxցa f (x) and limxցa g(x) exists, then: (i) limxցa (f + g)(x) = limxցa f (x) + limxցa g(x) (ii) limxցa (f · g)(x) = (limxցa f (x)) · (limxցa g(x)) (iii) if , ∀x ∈ S, g(x) 6= 0 and limxցa g(x) 6= 0, then xցa f (x) limxցa ( fg )(x) = lim lim xցa g(x)

sandwich rule for limits of functions let S ∈ R, f, g, h : S → R. let a ∈ R such that (a, a + b) for some b > 0. if f (x) ≤ h(x) ≤ g (x) in (a, a + b) and limxցa f (x) = limxցa g(x) = r then limxցa h(x) = r

propositions and friends 3.2.3: let S ∈ R, f, g : S → R be continuous then f + g : S → R, f · g : S → R and gf : S → R are continuous 3.3.3: let f : [a, b] → R be a continuous and injective function. let f −1 : f ([a, b]) → [a, b] be the compositional inverse of f then: (i) f is strictly monotone (ii) f −1 is strictly monotone (precisely, f − 1 is strictly increasing if f is strictly increasing and vice versa) (iii) f −1 is continuous 3.4.2: all general statements regarding limits from above also hold for translated version from below 3.4.4: let S ⊆ R and let f : S → R. let a ∈ R be such that (a, a + h) ∈ S for some h > 0. let r ∈ R. define fˆ by fˆ = f (x) if a < x and fˆ = r is x = a. then lim f (x) = r ⇐⇒ fˆ is continuous at a

xցa

3.4.5: let S ⊆ R and let f : S → R. let a ∈ R be such that (a, a + h) ∈ S for some h > 0. let r ∈ R. then limx→a f (x) = r ⇐⇒ for every (monotone) sequence (an )n∈N in (a, a + h) that converges to a, (f (an ))n∈N converges to r 3.4.6: let S ⊆ R and let f : S → R. let a ∈ R be such that (a − h, a + h) ∈ S for some h > 0. then f is continuous at a ⇐⇒ limx→a f (x) = f (a) 3.4.11: limx→∞ f (x) = limtց0 f ( 1t ) 3.4.13: limxցa f (x) = ∞ ⇐⇒ there is some h > 0 with 1 f (x) > 0 for all x ∈ (a, a + h) and limxցa f (x) =0

definition

and another theorem

differentiable at x0 : S open, f : S → R, x0 ∈ S then (x0 ) limx→x0 f (x)−f exists x−x0 differentiable: differentiable at each x0 in S (x0 ) derivative of f at x0 : f ′ (x0 ) = limx→x0 f (x)−f x−x0

let f, g : [a, b] → R be continuous functions which are differentiable in (a, b). then there is some ξ ∈ (a, b) with f (ξ ) · (g(b)g (a)) = g (ξ ) · (f (b)f (a))

local maximum: S ∈ R, f : S → R, x0 ∈ S then ∃h > 0 ∀x ∈ S (x0 − h < x < x0 + h ⇒ f (x0 ) ≥ f (x)) local minimum: S ∈ R, f : S → R, x0 ∈ S then ∃h > 0 ∀x ∈ S (x0 − h < x < x0 + h ⇒ f (x0 ) ≤ f (x)) local extremum: local minimum or local maximum

mean value theorem let f : [a, b] → R be a continuous function which is differentiable in (a, b), then there is some ξ ∈ (a, b) with f (b)−f (a) f ′ (ξ) = b−a

chain (composite) rule let f:S → R be differentiable at x0 , f (S) ⊆ T and let g : T → R be differentiable at f (x0 ). then also g ◦ f : S → R is differentiable at x0 and (g ◦ f )′ (x0 ) = g ′ (f (x0 )) · f ′ (x0 )

cauchy mean value theorem let f, g : [a, b] → R be a continuous functions which are differentiable in (a, b). then there is some ξ ∈ (a, b) with (b)−f (a) f ′ (ξ) = fg(b)−g(a) g ′ (ξ)

algebra of differentiable functions let f, g : S → R be differentiable at x0 , then: (i) (f + g)′ (x0 ) = f ′ (x0 ) + g ′ (x0 ) (ii) (f · g)′ (x0 ) = f ′ (x0 ) · g(x0 ) + f (x0 ) · g ′ (x0 ) (iii) if g(x) 6= 0 for all x ∈ S then g (x0 )f ′ (x0 ) − g ′ (x0 )f (x0 ) f ( )′ (x0 ) = g g 2 (x0 )

l’hopital’s rule let f, g : [a, b) → R be continuous functions which are differentiable in (a, b). suppose (i) g ′ (x) 6= 0 for all x ∈ (a, b) (ii) f (a) = g(a) = 0 f ′ (x) (iii) limxցa g ′ (x) exists then f (x) f ′ (x) = lim ′ lim xցa g(x) xցa g (x)

some theorem let f : S → R be a continuous function and differentiable at x0 ∈ S. if f is injective with f ′ (x0 ) 6= 0, then f − 1 defined on T = f (s) is differentiable at y0 := f (x0 ) and (f −1 )′ (y0 ) =

1 f ′ (f −1 (y0 ))

some other theorem if f : S → R is differentiable at x0 ∈ S and x0 is a local extremum of f , then f ′ (x0 ) = 0

propositions and friends 4.1.2: if f : S → R is differentiable at x0 ∈ S, then f is continuous at x0 4.1.3: let f : S → R be differentiable at x0 , then (i) if f ′ (x0 ) > 0, then these is some h > 0 such that for all x1 , x2 ∈ (x0 − h, x0 + h), we have x1 < x0 < x2 ⇒ f (x1 ) < f (x0 ) < f (x2 ) (ii) if f ′ (x0 ) < 0, then these is some h > 0 such that for all x1 , x2 ∈ (x0 − h, x0 + h), we have x1 < x0 < x2 ⇒ f (x1 ) > f (x0 ) > f (x2 )

rolle’s theorem let f : [a, b] → R be a continuous function which is differentiable in (a, b). if f (a) = f (b), then there is some ξ ∈ (a, b) with f ′ (ξ) = 0.

4.2.2: let S, T ⊆ R be open intervals and let f : S → T, g : T → R be functions. if f and g are differentiable, then also g ◦ f : S → R is differentiable and (g ◦ f )′ = (g ′ ◦ f ) · f ′...