Midterm 1 Study Aid

Topics:

· Algebraic properties (rules of arithmetic)
· rationals and irrationals
· positive/negative and order properties
· completeness of the real numbers

· bounds and Supremum/Infimum
· Completeness property

· Triangle inequality
· Archimedean property of N
· Density of Q in R
· Uncountability of R

Chapter 3: Sequences

Topics:

· Sequences
· Limit theorems
· Monotone sequences
· Subsequences
· Cauchy criterion and Cauchy sequences

Definitions:

· Limits, i.e. lim(x_n)=X
· Tail of a sequence
· Monotonicity and boundedness
· Subsequence
· Cauchy criterion
 

· Uniqueness of a limit
· lim(x_n+ y_n)=lim(x_n)+lim(y_n) etc.
· Squeeze theorem
· Ratio test (for positive sequences)
· Monotone convergence theorem
· Bolzano-Weierstrass
· Cauchy = convergent