Find a limit involving floor function

I had to find the following limit: $displaystyle{lim_{x to infty}}frac{x}{lfloor x rfloor}$ Where $xinmathbb{R}$ and $f(x)= {lfloor x rfloor}$ denotes the floor function.

This is what I did:

I wrote $x-1leq lfloor x rfloorleq x$. Then $frac{1}{x}leqfrac{1}{lfloor x rfloor}leqfrac{1}{x-1}$. Multiplying by $x$ we get $frac{x}{x}leqfrac{x}{lfloor x rfloor}leqfrac{x}{x-1}$. Since $displaystyle{lim_{x to infty}}frac{x}{x}=displaystyle{lim_{x to infty}}frac{x}{x-1}=1$, by the squeeze theorem we can say that $displaystyle{lim_{x to infty}}frac{x}{lfloor x rfloor}=1$. Is this correct? If not, what should I fix?