prove that $\lfloor x\rfloor\lfloor y\rfloor\le\lfloor xy\rfloor$

First note that $f:\mathbb R\rightarrow \mathbb Z$ given by $f(x)=\lfloor x\rfloor$ is an increasing function that is the identity on the integers. Then note that for positive $x$ we have $0\leq f(x)\leq x$. With this we get $$ f(x)f(y)\leq xy $$ and applying the increasing function $f$ on both sides above noting that the left hand side is an integer we then get: $$ f(x)f(y)=f(f(x)f(y))\leq f(xy) $$ which proves the claim.

Assuming that you mean $\lfloor x \rfloor \lfloor \color{red}{y} \rfloor \le \lfloor xy \rfloor$, write

$$x := a + b, \quad y = c + d, \quad a,c \in \mathbb{N} \cup \{0\}, \quad b,d \in [0,1\rangle.$$

Then $\lfloor x \rfloor \lfloor y \rfloor = ac$ and

$$\lfloor xy \rfloor = \lfloor (a+b)(c+d) \rfloor = \dots$$

Is every projection continuous?

Prove that $M$ is a free module if and only if $M$ is a projective module over $PID$.

If a sub-C*-algebra does not contain the unit, is it contained in a proper ideal?

Order of Double Coset

Mathematical way of determining whether a number is an integer

$ \int\frac{g(x)}{f(x)} \, dx $ where $f(x) = \frac{1}{2}(e^x+e^{-x})$ and $g(x) = \frac{1}{2}(e^x - e^{-x})$?

Gamma & Zeta Summation $\sum_{n=0}^{\infty}\frac{\Gamma(n+s)\zeta(n+s)}{(n+1)!}=0$

Let (X,Y) have a Dirichlet Distribution with paramters $(\alpha_1, \alpha_2, \alpha_3)$ Establish that X~Beta$(\alpha_1, \alpha_2 + \alpha_3)$ [closed]