Decide which of the following are true statements. Provide a short justification

Question

Decide which of the following are true statements. Provide a short justification for those that are valid and a counterexample for those that are not (any answer without a justification or counterex- ample will not be considered): 1. Since the Rational numbers are a subset of the Real numbers the Axiom of Completeness also applies to the Rational numbers. In general, the order of the terms in infinite series it is not important since it has infinite terms. 2. f any two sets always contains more elements than their intersection. 4 If supA

Explanation / Answer

1.This statement is false.consider the set of rationals constructed from the partial sums for the Taylor's expansion for e (the base of natural logs). Can you see why the set is rational and bounded, but has no LUB?

2.This statement is false.counter example is

11/2+1/31/4+1/5+=0.693147

But if we rearrange the terms as follows the value of the series gets influenced by this action:

1+1/31/2+1/5+1/71/4+=1.03972

3.This statement is also false as if set A=set B then

number of elements in union=number of elements in intersection.

4.this statement is true.

5.This statement is false

No. For example, take the sequence

an={0 if n is even

1 if n is odd}

It is bounded because it stays inside the interval [0,1], but it has no limit.

Intuitively, you shouldn't expect that bounded convergent, because even if the terms of a sequence stay in some general area, doesn't mean that all of its terms must always be getting closer and closer to each other (which is what the notion of Cauchy sequence captures; a sequence in RR or CC is convergent it is Cauchy).

So commutativity of addition isn't true on infinity? How was it obtained and ho

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