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I have a problem with this excercise. I need your help.

Let $f:\mathbb{R}\longrightarrow\mathbb{R}$

$f(t)=t+[t]$

where $[\cdot]$ is the floor function.

Define the metric:

$$d(x, y)=|f(x)-f(y)|\quad (x,y)\in\mathbb{R}^2$$

  1. The metric space $(\mathbb{R}, d)$ is complete?

  2. Can $(\mathbb{R}, d)$ be expressed as countable union of its compact subset?

For 1. I think I have to show that a Cauchy sequence is convergent on $(\mathbb{R}, d)$ right?

For 2. I have no idea.

Help me please :)

  • 0
    For 2. you could try to determine $(\mathbb R, d)$'s compact subsets ...2012-04-14
  • 0
    For 1., you could try $x_n=1-1/n$. Do you think the sequence $(x_n)$ is Cauchy? (Yes.) Does it converge?2012-04-14
  • 0
    For 2., you could try to show that $[k,k+n/(n+1)]$ is compact, for every integers $n\geqslant1$ and $k$.2012-04-14
  • 0
    This is a typical case where you could write a full solution to your own question.2012-04-14

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