Question

5. Show that a non empty interval (a, b) of R contains an infinite set of rational numbers and an infinite set of irrational num- bers. Hint: 1) Show that neither ?2 , nor q?2 with q ? Q are rationnal. 2) Show that there exists n ? N st. a + ?2/n ? (a, b) and st. a + 1/n ? (a, b). 3) Show that for well chosen p, q ? Z, one has p/2n ? (a, a + 1/n) and q/2n ? (a, a + ?2/n).

          5. Show that a non empty interval (a, b) of R contains
an infinite set of rational numbers and an infinite set of irrational num-
bers.

Hint:
1) Show that neither ?2 , nor q?2 with q ? Q are rationnal.
2) Show that there exists n ? N st. a + ?2/n ? (a, b) and st. a + 1/n ? (a, b).
3) Show that for well chosen p, q ? Z, one has p/2n ? (a, a + 1/n) and
q/2n ? (a, a + ?2/n).

5. Show that a non empty interval (a, b) of R contains
an infinite set of rational numbers and an infinite set of irrational num-
bers.

Hint:
1) Show that neither ?2 , nor q?2 with q ? Q are rationnal.
2) Show that there exists n ? N st. a + ?2/n ? (a, b) and st. a + 1/n ? (a, b).
3) Show that for well chosen p, q ? Z, one has p/2n ? (a, a + 1/n) and
q/2n ? (a, a + ?2/n).

Added by Michael K.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

01/21/2023

Step 1

To do this, we use the fact that q = V2 - r and the inequality V2 < rq. This shows that q € Q and so q is a rational number. 2) To show that there exists n € N such that st. a + V e (a,6) and st. a + 4 € (a,b) exist, we need to show that there exists a rational Show more…

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Show that a non empty interval (a, b) of R contains an infinite set of rational numbers and an infinite set of irrational numbers. Hint: 1) Show that neither −2 , nor q−2 with q ∈ Q are rationnal. 2) Show that there exists n ∈ N st. a + −2/n ∈ (a, b) and st. a + 1/n ∈ (a, b). 3) Show that for well chosen p, q ∈ Z, one has p/2n ∈ (a, a + 1/n) and q/2n ∈ (a, a + −2/n).