Real Analysis/Advanced Calculus (Needs to be a full proof), Mathematics

Both need to be a full page, detailed proof. Not just a few lines of proof.

(1) “Every convergent sequence contains either an increasing, or a decreasing subsequence (or possibly both, of course).”
Proof: Suppose that lim(as n goes to infinity) x(sub n) = L. If infinitely many terms are equal to L, we are done. If not, either infinitely many terms are > L, or infinitely many terms are < L. Choose one option and complete the proof.

(2) Adapt carefully the bisection argument seen in the proof of Bolzano-Weierstrass to prove the Intermediate Value Theorem: If f is a continuous function on [a,b], and if f(a) and f(b) have opposite signs, then there exists a point c (a,b) such that f(c) = 0.
Proof: Suppose that f(a) < 0 and f(b) > 0. Denote a(sub 0) = a, b(sub 0) = b and consider the intervals [a, (a+b)/2] and [(a+b)/2, b].
Posted Date: 3/13/2013 9:18:42 PM | Location :







Related Discussions:- Real Analysis/Advanced Calculus (Needs to be a full proof), Assignment Help, Ask Question on Real Analysis/Advanced Calculus (Needs to be a full proof), Get Answer, Expert's Help, Real Analysis/Advanced Calculus (Needs to be a full proof) Discussions

Write discussion on Real Analysis/Advanced Calculus (Needs to be a full proof)
Your posts are moderated
Related Questions

A radioactive substance decays to 30% of its original mass in 15 months. Determine the half-life of this radioactive substance to the nearest month

If the minute hand of a big clock is 1.05 m long, find the rate at which its tip is moving in cm per minute.

who ,why and when discovered unitary method

Determine the equation of the tangent line to r = 3 + 8 sinθ at θ = Π/6. Solution We'll first need the subsequent derivative. dr/dθ = 8 cosθ The formula for the deriv

PLAY AND LEARN :  Children can learn many basic mathematical concepts through games. They enjoy Mathematical concepts can be playing within familiar contexts. Their games also gen

The temperature at 6 P.M. was 31°F. Through midnight, it had dropped 40°F. What was the temperature at midnight? Visualize a number line. The drop from 31° to 0° is 31°. There

Q. Show Inverse Trigonometric Functions? Ans. Many functions, including trig functions, are invertible. The inverse of trig functions are called ‘inverse trig functions'.

What is Inductive Reasoning ? Sometimes we draw conclusions based on our observations. If we observe the same results again and again, we conclude that the event always has the

Which of the subsequent terms does NOT describe the number 9? Nine is NOT prime since it has 3 factors; 1, 3, and 9. Prime numbers have only 2 factors.