Least Upper Bound Property

An ordered set S is said to have the least-upper-bound property if:

for E < S, E not empty, and E bounded above, then

Suppose S is an ordered set with the LUB property, B<S, B not empty, and B bounded below. Then

1) α = sup L ____

2) α = 

3) In particular, inf B ____

1) exists in S 

2) inf B

3) exists in S

An ordered field is a field F which is also an ordered set, such that:

1) x+y < x+z if ___

2) xy > 0 if ___

1) x,y,z ∈ F and y < z

2) x ∈ F, y ∈ F, x > 0, and y > 0

Archimedian Property

If x ∈ R , y ∈ R, and x > 0, then

there exists a positive integer n such that nx > y

Q is dense in R 

 If x ∈ R , y ∈ R, and x < y, then

1) If X is a metric space with E < X, and if E' denotes the set of all limit points of E in X, then the closure of E is___

2) Ē is ___

3) E = Ē iff

4) Ē < F for every

1) the set Ē = E ∪ E'.

2) closed.

3) E is closed.

4) closed set F < X such that E < F.

Heine-Borel

For a set E in R^k, the following properties are equivalent:

1) E is closed and

2) E is

3) Every infinite subset of E has

1) bounded.

2) compact

3) a limit point in E.

Weierstrass

Every bounded infinite subset of R^k has

1) Two subsets A and B of a metric space X are said to be separated if 

2) A set E < X is said to be connected if E is *not*

1) both A ∩ cl(B) and cl(A) ∩ B are empty; i.e., if no point of A lies in the closure B and no point of B lies in the closure of A.

2) a union of two nonempty separated sets.

1) A sequence {f_n} in a metric space X is said to converge if

2) In this case, lim_n→∞ =

1) there exists a point p ∈ X such that for every ε > 0 there exists an integer N such that n ≥ N implies d(f_n, f_m) < ε.

2) p

Let {f_n} be a sequence in a metric space X.

a) {f_n} converges to p ∈ X iff 

b) If p ∈ X, p' ∈ X, and if {f_n} converges to p and p', then

c) If {f_n} converges, then 

d) If E < X and if p is a limit point of E, then

a) every neighborhood of p contains f_n for all but finitely many n.

b) p'= p.

c) {f_n} is bounded.

d) there is a sequence {f_n} in E such that p = lim_n→∞ f_n.

1) Given a sequence {f_n}, consider  sequence {n_k} of positive integers, such that n₁ < n₂ < ... Then the sequence {f_ni} is called a

2) If {f_ni} converges, its limit is called a

1) subsequence of {f_n}.

2) subsequential limit of {f_n}.

a) If {f_n} is a sequence in a compact metric space X, then

b) Every bounded sequence in R^k contains

a) some subsequence of {f_n} converges to a point of X.

b) a convergent subsequence.

A sequence {f_n} in a metric space X is said to be a Cauchy sequence if

1) Given a set E in a metric space X, diam E =

2) If K is a sequence of compact sets in X such that K_n > K_n+1 (n=1,2,3,...), and if lim_n→∞ K_n = 0, then

1) diam cl(E).

2) ∩^∞₁ K_n consists of exactly one point.

a) In any metric space X, every convergent sequence is a 

b) If X is a compact metric space, and if {f_n} is a Cauchy sequence in X, then

c) In R^k, every Cauchy sequence

a) Cauchy sequence.

b) {f_n} converges to some point in X.

c) converges.

a) If p > 0, then lim_n→∞ 1/n^p = 

b) If p > 0, then lim_n→∞ ⁿ√p =

c) lim_n→∞ ⁿ√n =

d) If p > 0 and a is real, then lim_n→∞ n^a/(1+p)ⁿ =

e) If |x| < 1, then lim_n→∞ x^_n =

a) 0

b) 1

c) 1

d) 0

e) 0

a) If |a_n| ≤ c_n for n ≥ N₀, where N₀ is some fixed integer, and if Σc_n converges, then 

b) If a_n ≥ d_n ≥ 0 for n ≥ N₀, and if Σd_n diverges, then

a) Σa_n converges.

b) Σa_n diverges.

a) If |a_n| ≤ c_n for n ≥ N₀, where N₀ is some fixed integer, and if Σc_n converges, then 

b) If a_n ≥ d_n ≥ 0 for n ≥ N₀, and if Σd_n diverges, then

a) Σa_n converges.

b) Σa_n diverges.

1. If 0≤x<1, then Σ^∞_{n=0 x}ⁿ =

2. If x≥1, then the series

1. 1/(1-x)

2. diverges.

1. Σ 1/n^p converges if

2. and diverges if

1. p > 1.

2. p ≤ 1.

1. If p > 1, then Σ^∞_n=2 1/(nlogn)^p___

2. If p ≤ 1, then

1. converges

2. the series diverges

Root Test

Given Σa_n , define L = lim_n→∞ |a_n|^1/n

a) if L < 1, then Σa_n ___

b) if L > 1, then Σa_n ___

c) if L = 1, then___

a) converges

b) diverges

c) the test gives no information.

Ratio Test

Given the series Σa_n, define  [image]. Σa_n

a) Converges if 

b) Diverges if 

a) L < 1

b) |(a_n+1)/a_n| ≥ 1 for all n ≥ n₀, where n₀ is some fixed integer, or L > 1

Given the power series Σ^∞_n=0 c_n(x-a)ⁿ about a, define  α= lim_n→∞ sup ⁿ√|c_n|, R = 1/α, 

a) If α = 0, R = 

b) If α = +∞, R =

c) Σ^∞_n=0 c_n(x-a)ⁿ converges if

d) Σ^∞_n=0 c_n(x-a)ⁿ diverges if 

a) +∞

b) 0

c) |x - a| < R

d)  |x - a| > R

Given two sequences {a_n} and {b_n}, define

A_n = Σⁿ_k=0  a_k if n ≥ 0; and define A_-1 = 0. Then, if 0 ≤ p ≤ q, we have Σ^q_n=p a_nb_n =

Σa_nb_n converges if

a) the partial sums A_n of Σa_n ___

b) b₀ ___

c) lim_n→∞ bn = ___

a) form a bounded sequence

b) ≥ b₁ ≥ b₂ ≥ ...

c) 0