Question

+ In Exercise 11 from Tutorial 6, we showed that if is an irrational number and y is a nonzero rational number, then ry is an irrational number. For example, 23 and are both irrational In Tutorial 5, we proved that between any two real numbers, there is a rational number. In Tutorial 6. we showed that the set of irrational mumbers is an uncountable set, which means that in some sense) there are many more irrational munber than there are rational numbers. It should come as no surprise to you that between any two real numbers, there is also an irrational number Exercise 5: Let and be real numbers such that < Show that there is an irrational number such that arc (Hint: Follow the Breakdown from Tutorials, but definek to be the largest integer such that V2+SA) We will stay in Section 2 2 of the book. Again, you are responsible for evercises 1-12 from section 22, but you only need to write up the ones on the following pages Book Exercise 7: Show that to the usual topology) finer than the finite complement topology for R (See Exercises from section 2.1) In 11 and 12. if you determine that the function is continuous you do not have to justify your answer, but if you determine that it is not continuous, then you must give an example of an open set in the codomain who invete image is not open in the domain (List both sets) is Book Exercise 11: Determine with justification) if the function / RR given by fli) - 1 -2, il (a) T-Tec (el Tv - Ti continuous (b) T - Tir continuo (f) Te - continuous

With exercise 5, the first person did it wrongly. We are to define k to be the largest integer such that root 2+k/n is less than or equal to a. Please an expert should solve this

Answer 1

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Exercise 5 let a and b be real numbers & axb. there is irrational number n such an show that that alakb. Soln There is an to do this easy way By Tutorial and b are also real numbers T2 2 which 9 Lb 12 12 5, 3 a rational number or such that qara b R 2 n=852 Then take Hence, we see that Where that aLaab m=812 is an irrational number