Definitions Quiz 3-28

Question 1

Bolzano-Weierstrass Theorem (Star theorem)

Answer 1

Every Bounded real sequence has a convergent subsequence

Question 2

Monotone Convergence Theorem

Answer 2

Every Bounded monotone sequence converges

Question 3

Cauchy Sequences

Answer 3

if xn is cauchy, then for every e>0, there is an N (natural number) s.t. for all m,n >= N, then |xn - xm| < e (ALL Cauchy sequences are convergent!)

Question 4

Two-Sided Limit Convergence

Answer 4

Let I be an open interval, a is a point in I, and the function f be defined everywhere on I except possibly at a.
We say that f(x) converges to L as x->a IFF given e>0, there exists d> 0 s.t. for all points x in I, with 0< |x-a| <d we have |f(x) - L| < e

Question 5

Two-sided Limit comparison

Answer 5

Let I be an open interval, a is a point in I, and the functions f,g be defined everywhere on I except possibly at a.
If f(x) = g(x) for all points x in I except a, then lim f(x) = lim g(x)

Question 6

Sequential Characterization of Limits

Answer 6

Let I be an open interval, a is a point in I, and the function f be defined everywhere on I except possibly at a.
Then lim f(x) = L IFF lim f(xn) = L for all sequences {xn} with xn in I except at a, s.t. lim xn = a.