2
$\begingroup$

Consider the metric space $\langle \mathbb I,d\rangle$ where $\mathbb I$ is the set of all irrational numbers, and $d$ is the usual distance metric. For each $n\in\mathbb Z^+$, let $x_n =\frac{n + \sqrt 2}{n-\sqrt 2}$. Then:

a. Prove that $\langle x_n\rangle$ is a sequence in $\mathbb I$.

b. As a sequence in $\langle \mathbb I,d\rangle$, prove that $\langle x_n\rangle$ does not converge.

Please help us answer this problem...

  • 0
    Hello @Bok and welcome to Math StackExchange! It's best if you let us know what you've tried so far so we can help you. What exactly are you stuck on?2012-12-06
  • 0
    I notice you originally wrote "Q" and said "irrational," but from context it's clear that you meant "irrational." Just be alert in the future that Q is usually used for exactly the opposite of what you meant *the rationals*.2012-12-06
  • 0
    a) For each n, times the numerator and denominator of x_n by (n + sqrt(2)) and see what happens.2012-12-06
  • 0
    b) Hint: What is the approximate value of x_n when n is very large e.g. 100000 ?2012-12-06

1 Answers 1