3. Basic Results about R^n

Question 1

Proposition 3.7
Uniqueness of Limits

Question 2

Proposition 3.8
Componentwise convergence
Proof

Question 3

Lemma 3.15
(x_j) -> x => …

Question 4

Proposition 3.14
Convergent sequences are bounded.

Question 5

Proposition 3.17
Completeness of R^n

Question 6

Theorem 3.18
Bolzano-Weierstrass for bounded sequences of vectors.

Question 7

Definition 3.20
Sequential continuity

Question 8

Definition 3.22
Continuous limit

Question 9

Definition 3.23
Separately continuous

Question 10

Exercise 3.5
Continuity => …

Question 11

Exercise 3.4
\epsilon-\delta continuity equiv.

Question 12

Propositon 3.29
f : U -> R^k is continious at p \in U iff.

Question 13

Lemma 3.31
R^(n+l) = R^n (+) R^l = {…}.
Denote projections. then…

Question 14

Proposition 3.32
E \subset R, a \in R and g : E -> R, define projections….

Question 15

Example of continuous rational functions (sort of)

Question 16

Definition 3.33
continuous along lines.

Question 17

Example 3.34
Separately continuous but not continuous along lines.

Question 18

Example 3.37
Continuous along lines at (0,0) but not continuous everywhere

Question 19

Example 3.38
continuous alog lines at every point, but not continuous.