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Problem 3: (Practice With Bounded Definitions)

Consider the set $S=[0,4) = \{x\in\mathbb{R}\mid 0\leq x \text{ and } x< 4\}$.

  1. Show that $S$ is bounded below by giving a lower bound. Prove that the number you gave is a lower bound, and then state another lower bound different than the one you gave.
  2. Of all possible lower bounds, which is the greatest lower bound. In other words, produce a lower bound $m$ so that if $m'$ is any lower bound, then we must have $m'\leq m$. Prove your answer.
  3. Show that $S$ is bounded above by giving an upper bound. Justify your answer.
  4. Of all possible upper bounds, which is the least upper bound?

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