Chapter 1 - Theorems

Question 1

What is the monotone convergence theorem?

Answer 1

If (a_n) is bounded and monotonic then (a_n) converges

Question 2

What is the bolzano-weierstrass theorem?

Answer 2

Every bounded real sequence has a convergent subsequence

Question 3

What is the theorem for if a sequence is Cauchy?

Answer 3

A real sequence converges if and only if it is Cauchy

Question 4

What is the theorem for uniqueness of limits?

Answer 4

If f has a limit at a, this limit is unique

Question 5

What is the theorem for algebra of limits?

Answer 5

Let a be a cluster point of D and f goes from D to the real numbers, and g goes from D to the real numbers, the limit x goes to a of f(x)=L and the limits of x goes to as of g(x)=K. Then,
1. the limit x goes to a of (f(x)+g(x)) = L+K
2. the limit x goes to a of f(x)g(x)=LK
3. the limit x goes to a 1/f(x) =1/L if 0 is not in f(0) and L does not equal 0

Question 6

What is the theorem for