Let x,y ∈ R satisfy
x < y. Prove that there exists a q ∈
Q such that x < q <
y.
Strategy for solving the problem
Show that there exists an n ∈
N+ such that 0 < 1/n <
y - x.
Letting A = {k : Z |
k < ny}, where Z denotes the
set of all integers, show that A is a non-empty subset of
R with an upper bound in R.
(Hint: Use...
Let {an} be a bounded sequence. In this question,
you will prove that there exists a convergent subsequence.
Define a crest of the sequence to be a
term am that is greater than all subsequent terms. That is,
am > an for all n > m
(a) Suppose {an} has infinitely many crests. Prove
that the crests form a convergent subsequence.
(b) Suppose {an} has only finitely many crests. Let
an1 be a term with no subsequent crests. Construct a...
Let S ⊆ R and let G be an arbitrary isometry of R . Prove that
the symmetry group of G(S) is isomorphic to the symmetry group of
S. Hint: If F is a symmetry of S, what is the corresponding
symmetry of G(S)?
a. Let A be a square matrix with integer entries.
Prove that if lambda is a rational eigenvalue of A then in fact
lambda is an integer.
b. Prove that the characteristic polynomial of the
companion matrix of a monic polynomial f(t) equals f(t).
Let f : R → R be a function.
(a) Prove that f is continuous on R if and only if, for every
open set U ⊆ R, the preimage f −1 (U) = {x ∈ R : f(x) ∈ U} is
open.
(b) Use part (a) to prove that if f is continuous on R, its zero
set Z(f) = {x ∈ R : f(x) = 0} is closed.
Let t be a positive integer. Prove that, if there exists a
Steiner triple system of index 1 having v varieties, then there
exists a Steiner triple system having v^t varieties
Let P be a partial order on a finite set X. Prove that there
exists a linear order L on X such that P ⊆ L. (Hint: Use the proof
of the Hasse Diagram Theorem.) Do not use induction