Topic 6 + 7 - Real Numbers & Functions

Question 1

Nested Interval Theorem

Answer 1

Let (a_n) and (b_n) be real sequences such that (a_n) ≤ (b_n) and [a_{n+1}, b_{n+1}] ⊆ [a_n, b_n]. Then the intersection of all intervals is non-empty.

⋂ [a_n,b_n] ≠ ∅
n∈N

Question 2

Sequential continuous at x_0

Answer 2

A function f: I → IR is sequentially continuous at the point x_0 if for every sequence (x_n) where lim x_n = x_0, lim f(x_n) = f(x_0).

Question 3

Sequentially continuous function

Answer 3

A function sequentially continuous at all points in I (x_0 ⊆ I), implying that lim f(x_n) = f (lim (x_n)) for all convergent (x_n).

Question 4

Intermediate Value Theorem

Answer 4

Let a function f be sequentially continuous and have a domain which is a closed interval.
Take a h such that f(a) ≤ h ≤ f(b), assuming f(a) ≤ f(b).
Then there exists a c ∈ [a,b] such that f(c) = h.

Question 5

Exponential function

Answer 5

exp: IR → IR
∞
exp (x) = ∑ x^{n} / n!
n=0

Question 6

Define Interval

Answer 6

I ⊆ R is an interval iff ∀ y1, y2 ∈ I, with y1 ≤ y2, then ∀y such that y1 ≤ y ≤ y2 implies y ∈ I.