If $X$ is an exponential random variable, does that mean $\lfloor X\rfloor$ and $X -\lfloor X\rfloor$ are independent?

Question

I've derived an expression for $P\{\lfloor X\rfloor= n, X -\lfloor X\rfloor\leq x\}$ to be $e^{-\lambda(n)}[1 - e^{-\lambda(x)}]$,

and when I find an expression for $P(\lfloor X\rfloor= n)P(X \leq x+n)$,

I have the two expressions aren't equal. Is this enough to say they aren't independent?