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I stumble upon this problem which says:

The set $\{ x\in\mathbb R: x\sin x\le 1, x\cos x \le 1\}$ is contained in $\mathbb R$.Then which of the following about the set is true:

  1. a bounded closed set
  2. a bounded open set
  3. an unbounded closed set
  4. an unbounded open set.

Any kind of hints will be helpful.

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    The inverse image of a closed set under a continuous function is closed.2012-11-19

1 Answers 1

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The inverse image of a closed set under a continuous function is closed.

For $x$ positive, just look at intervals where the sine and cosine are both negative, and those would be included, so the set is unbounded.

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    Let A={x€R: x sinx <=1} and B={x€R; x cosx<=1}.Now if we choose x to lie in the interval (pi+2n* pi,3*pi/2+2n*pi),then we see that both A and B lie in (-∞,1] as for those values of x which lie in the aforementioned interval,both sinx and cosx will be negative.Now we see that both A and B are closed (semi-closed ). Since intersection of two closed sets is closed ,we can conclude that the set must be closed and unbounded as well.2012-11-20