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Prove or disprove each of these statements about the floor and ceiling functions.

  1. $\left \lfloor \left \lceil x \right \rceil \right \rfloor = \left \lceil x \right \rceil$ for all real numbers $x.$
  2. $\left \lfloor x+y \right \rfloor = \left \lfloor x \right \rfloor+\left \lfloor y \right \rfloor$ for all real numbers $x.$
  3. $\left \lceil \left \lceil x/2 \right \rceil /2\right \rceil \left \lceil x/4 \right \rceil$ for all real numbers $x.$
  4. $\left \lfloor  \left \lceil x \right \rceil^{-1/2} \right \rfloor$  =$\left \lfloor x \right \rfloor^{-1/2}$ for all positive real numbers $x.$
  5. $\left \lfloor x \right \rfloor+\left \lfloor y \right \rfloor +\left \lfloor x+y \right \rfloor <= \left \lfloor 2x \right \rfloor=\left \lfloor 2y \right \rfloor$ for all real numbers $x$ and $y.$
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