实数上取整性质证明：若⌈x⌉−x≥1/2，求证⌈2x⌉=2⌈x⌉−1

Let's work through this problem systematically—we just need to lean on the formal definition of the ceiling function to make the connection clear.

Step 1: Recall the ceiling function definition #

For any real number $x$, $\lceil x \rceil$ is defined as the smallest integer greater than or equal to $x$. Mathematically, if we let $n = \lceil x \rceil$ (where $n$ is an integer), this means:

$n - 1 < x \leq n$

Step 2: Apply the given condition #

The problem states $\lceil x \rceil - x \geq \frac{1}{2}$. Substituting $n = \lceil x \rceil$, this translates to:
$$n - x \geq \frac{1}{2}$$
Rearranging gives:

$x \leq n - \frac{1}{2}$

Step 3: Narrow down the range of $x$ #

Combine the definition from Step 1 and the condition from Step 2 to get the full range of $x$:

$n - 1 < x \leq n - \frac{1}{2}$

Step 4: Analyze $2x$ and its ceiling #

Multiply every part of the inequality by 2 to find the range of $2x$:
$$2(n - 1) < 2x \leq 2\left(n - \frac{1}{2}\right)$$
Simplify both sides:

$2n - 2 < 2x \leq 2n - 1$

Now, let's look at $\lceil 2x \rceil$. By definition, this is the smallest integer greater than or equal to $2x$. From the inequality above:

• $2x$ is strictly greater than $2n - 2$ (which is $(2n - 1) - 1$)
• $2x$ is less than or equal to $2n - 1$, which is an integer

The smallest integer that satisfies being greater than or equal to $2x$ here is exactly $2n - 1$.

Step 5: Compare to the right-hand side #

The right-hand side of the equation we need to prove is $2\lceil x \rceil - 1$. Since $\lceil x \rceil = n$, this becomes:
$$2n - 1$$

Which matches the value we found for $\lceil 2x \rceil$. So the equality holds.

Quick check for edge cases #

What if $x$ is an integer? If $x$ is integer, $\lceil x \rceil - x = 0$, which violates the condition $\lceil x \rceil - x \geq \frac{1}{2}$, so these cases aren't part of the problem's premise and don't affect our proof.

内容的提问来源于stack exchange，提问作者Rick