Someone claims the following is true a b R a b 11 a211 b2

Someone claims the following is true: a, b R, a < b = 1/(1 +a^2)>1/(1 + b^2)

(a) What is the negation of the claim?

(b) Prove the claim is false by proving the negation is true.

(c) What is the largest open interval I such that the following corrected statement is true? “a, b I, a < b =1/(1 + a^)2>1/(1 + b^2) .” (Simplystate your result; proofs follow later. Note that to prove one’s answer is correct there are 2 things that need to be shown: first, that the statement is actually true on I; second, that I is actually the biggest such interval, by which we mean we can’t expand I tosomething bigger and still have the statement be true. This is the content of the next 2 parts.)

(d) Prove that the corrected statement is true for the I found in part(c).

(e) Prove by contradiction that I is the largest such open interval. For this we suppose that the statement is true for some even bigger open interval, i.e., for some J that properly contains I; then we reason to obtain a contradiction. Your proof should be structured as follows: “Suppose there were an open interval J such that I J, I 6= J, and that

a, b J, a < b = 1 /(1 + a^ 2) > 1/( 1 + b^ 2) . Insert valid reasoning leading to a contradiction. This contradiction shows that no such J exists, and so I is the largest interval for which the statement is true.”

Real analysis problem, please help,thank you.

Solution

Negation is:

if a> b, then

1/(1+a^2) < 1/(1+b^2)

If a > b,

then 1/1+a^2 will be less than 1/(1+b^2)