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
How to proof:
Matrix A have a size of m×n, and the rank is r. How can we
rigorous proof that the dimension of column space are always equal
to the dimensional of row space?
(I can use many examples to show this work, but how to proof
rigorously?)