<s>[INST] $\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers of $x$ and $y$ Give a convincing argument that $\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers $x$ and $y$. Could someone please explain how to prove this? I attempted to say that the largest values that could be added to $x$ and $y$ is $0.99$ and that doing so still made $\lfloor x\rfloor + \lfloor y\rfloor < \lfloor x+y\rfloor$. However, my answer was not accepted.
 [/INST] Let $x = a + m$ and $y = b + n$ where $a, b$ are integers and $m,n$ are real numbers such that $0 \le m,n < 1$. Then,
$$\lfloor x \rfloor + \lfloor y \rfloor = a + b$$
while
$$\begin{align}\lfloor x + y \rfloor &= \lfloor a + b + m + n \rfloor\\
&= a + b + \lfloor m + n \rfloor\\
& \ge a + b\\
&= \lfloor x \rfloor + \lfloor y \rfloor\end{align}$$
Equality occurs when the fractional part of $x$ and $y$ sum up to be less than $1$, in which case $\lfloor m + n \rfloor = 0$. For example, $x = 0.5$ and $y = 1.1$.
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