Question

18 A Course in Real Analysis 10. Prove that between any pair of real numbers a < b there exist infinitely many rational numbers and infinitely many irrational numbers.

Please answer question 10 and write legibly -thanks!

Answer 1

Betwein any pein of meaf nubers a6 thore exist infinitty many yoafionals numoirs and infinitely many imratmar numbers 1S4 Part Properkren of R, there exists a naturaf numben n eueh that (6-a)70 BY Archime dean nb-na> napl nb m such hnt Then an integer nat iR usig the Proper Hies o.f Arehimedean Porperty m- na Themefoe m nat < n b nakm SoMe have nazm no integer and nisa notua nunarr oatisnal mumbor is m am So mis a ket 9 m The n is such tha a9,b S0 bySimilarly Since etpa ,ta find . a rational mumeen ,e ,,), where ais realnumber and ,is rel mumlar and a<a Inthis fashin each n EN hve is a ontimal numlaem we obtain inductively fr

ifinitely S0 between a amd o hare ane LPRoved many roatiena) numkaens, 2nd Parat Ea, J2 b ase md 12a <V2 b 2uch Yoapinmat mumbe n Then Ja9 Ahat aく1くb 2 to and then Then is an irpHonal Let 8- numeen 8ahsfax usa the pranf sf p t Part we can eanily say that a and lo there are inimtely marmy ip tisnal numkors be tdesn