TAOCP 1.2.4 Exercise 2

For any real number $x$, the floor function $\lfloor x \rfloor$ is an integer by definition.

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Section 1.2.4: Integer Functions and Elementary Number Theory

Exercise 2. ▶ [01] What is $\lceil \lfloor x \rfloor \rceil$?

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Solve time: 27s

For any real number $x$, the floor function $\lfloor x \rfloor$ is an integer by definition. Applying the ceiling function to an integer does not change its value, since the ceiling of an integer is the integer itself. Therefore, we have

$\lceil \lfloor x \rfloor \rceil = \lfloor x \rfloor.$

This completes the proof. ∎