1.- Prove that the set of irrational numbers is uncountable by
using the Nested Intervals Property.
2.- Apply the definition of convergent sequence, Ratio Test or
Squeeze Theorem to prove that a given sequence converges.
3.- Use the Divergence Criterion for Sub-sequences to prove that
a given sequence does not converge.
Subject: Real Analysis
Prove the Converse of Proposition 3.3 by using Betweenness Axiom
1. The converse is Given B ? C ? D and A ? B ? D, then A ? B ? C
and A ? C ? D. Please do not use "by mapping of letters"
Use the Intermediate Value Theorem and the Mean Value Theorem to
prove that the equation cos (x) = -10x has exactly one real
root.
Not permitted to use words like "Nope", "Why?", or
"aerkewmwrt".
Will be glad if you can help me with this question, will
like to add some of your points to the one I have already summed
up.. Thanks
Prove the following theorem. Using the ruler function axiom. List
all axioms and definitions used.
Let P and Q be two points, then the line segment AB=BA (AB and
BA have lines over them to show line segments)
Use the Intermediate value theorem to find an interval of length
one that contains a root of the equation. a) x^3=9 b) 3x^3+x^2=x+5
2. Still consider equations 1a and 1b A)How many iterations are
required to find the roots by the Bisection Method within six
decimal places B) Use the interval you find in problem 1 that
contains a root to compute the first 4 iterations and table the
results
Prove that if f is a bounded function on a bounded interval
[a,b] and f is continuous except at finitely many points in [a,b],
then f is integrable on [a,b]. Hint: Use interval additivity, and
an induction argument on the number of discontinuities.