Find the limit of the function below

Question

I was solving the limit at zero of this function :

$f(x) = x(\frac{1}{x} -\large\lfloor{(\frac{1}{x})}\rfloor)$

See this image>1

I thought of the corollary of the sandwich theorem (which we use a lot in class)

As $x \to 0$

$(\frac{1}{x} -\large\lfloor{(\frac{1}{x})}\rfloor)$ and is bounded between 0 and 1

Then the function should tend to zero

Is my method correct?