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06082020realanalysis

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User Generated
Subject
Algebra
School
University of California Irvine
Type
Homework
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4.
Since rational number set Q is dense in R, for any rational number in [0,b], there is irrational numbers in
its any neighborhood; similarly, Irrational number is also dense in R, thus for any irrational number in
[0,b], there is at least one rational number in its neighborhood. Therefore
a) For any partition P = {=t
0
<t
1
…< t
n
=b}, we have
 


 


 

 


choose t
k
= kb/n, then


 

  

  
  

  


   

 


Its limit as n approaches infinity is b
3
n. Therefore,




b) Similarly,
 


 


   
 


Therefore




5. Since 

, here we assume c>0, (if c<0, will be -),

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4. Since rational number set Q is dense in R, for any rational number in [0,b], there is irrational numbers in its any neighborhood; similarly, Irrational number is also dense in R, thus for any irrational number in [0,b], there is at least one rational number in its neighborhood. Therefore a) For any partition P = {=t0 ...
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This is great! Exactly what I wanted.

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