Problem 3. This problem outlines one way to define an absolute value of a given

Question

Problem 3. This problem outlines one way to define an absolute value of a given real -1yl1S 11-vl. Hint: (b) Use (a) to show that if (an) is a Cauchy sequence then (lan) is also a Cauchy sequence number (a) Prove that for any x, y E Q the following inequality holds ll Use the triangle inequality (c) Use (a) to prove that if (an) and bn) are Cauchy sequences such that (an-bn) is a null sequence, then (lanbs also a null sequence (d) For x E R define its absolute value lll in the following way: 1 := (Pol), where (an) is some representative of r. Use (b) and (c) to show that is well-defined

Explanation / Answer

a)

Without loss of generality we assume

|x|<=|y|

So,

||x|-|y||=|y|-|x|

If x and y are the same sign then

|y|-|x|=|y-x|

Else

|y-x|=|y|+|x|>=|y|-|x|

HEnce proved

b)

Let, e>0 and N so that for all m,n>N

|am-an|<e

Using a)

||am|-|an||<=|am-an|<e

HEnce, (|an|) is also cauchy sequence

c)

Null sequence is a sequence with 0 as a limit

Hence,

||an|-|bn||<=|an-bn|

Since, an-bn is null sequence so for all e>0 there exist N so that for all n>N

|an-bn|<e

HEnce, ||an|-|bn||<e

Hence proved

d)

an ,bn are two representatives of x then an-bn is null sequence

Hence, |an|-|bn| is also null sequence

Ie |x| is unique hence well defined

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