Question

Prove that there is a positive real number *x* such that
x^{2} - 2 = 0.

What you'll need: Definition of a real number, definition of positive, definition of zero, and definition of Cauchy.

Answer #1

Suppose x is a positive real number such that x + (1/x) is an
integer. Prove that x^2019 + (1/x^2019) is an integer.

) Let α be a fixed positive real number, α > 0. For a
sequence {xn}, let x1 > √ α, and define x2, x3, x4, · · · by the
following recurrence relation xn+1 = 1 2 xn + α xn (a) Prove that
{xn} decreases monotonically (in other words, xn+1 − xn ≤ 0 for all
n). (b) Prove that {xn} is bounded from below. (Hint: use proof by
induction to show xn > √ α for all...

) Prove or disprove the statements: (a) If x is a real number
such that |x + 2| + |x| ≤ 1, then |x ^2 + 2x − 1| ≤ 2. (b) If x is
a real number such that |x + 2| + |x| ≤ 2, then |x^ 2 + 2x − 1| ≤
2. (c) If x is a real number such that |x + 2| + |x| ≤ 3, then |x
^2 + 2x − 1 |...

prove: Let the real number x have a Base 3 representation of: x
=
0.x0x1x2x3x4x5x6x7…
where xi is the ith digit of x.
Then, if xi ={ 0 , 2 } (or xi not equal to 1)
for all non-negative integers i, then x is in the Cantor set
(Cantor dust).
Think about induction.

Prove if on the real number line R , set A = 0, B = 1, X = x and
Y = y (for some x , y ∈ R ) then the condition that X , Y are
harmonic conjugates with respect to A , B (i.e. ( A , B ; X , Y ) =
− 1) means 1/x + 1/y = 2

Prove the following:
For any positive real numbers x and y, x+y ≥
√(xy)

1. Let x be a real number, and x > 1. Prove 1 < sqrt(x)
and sqrt(x) < x.
2. If x is an integer divisible by 4, and y is an integer that
is not, prove x + y is not divisible by 4.

Discreet Math: Prove or disprove each statement
a) For any real number x, the floor of 2x = 2 the floor of x
b) For any real number x, the floor of the ceiling of x = the
ceiling of x
c) For any real numbers x and y, the ceiling of x and the
ceiling of y = the ceiling of xy

Let λ be a positive irrational real number. If n is a positive
integer, choose by the Archimedean Property an integer k such that
kλ ≤ n < (k + 1)λ. Let φ(n) = n − kλ. Prove that the set of all
φ(n), n > 0, is dense in the interval [0, λ]. (Hint: Examine the
proof of the density of the rationals in the reals.)

Here are two statements about positive real numbers. Prove or
disprove each of the statements
∀x, ∃y with the property that xy < y2
∃x such that ∀y, xy < y2 .

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