Proof for $\left\lfloor\frac 1j\left\lfloor\frac nk\right\rfloor\right\rfloor=\left\lfloor\frac n{jk}\right\rfloor$ [duplicate]

Problem: For positive integers $n,j,k$, prove that the following holds:

$$\left\lfloor\frac 1j\left\lfloor\frac nk\right\rfloor\right\rfloor=\left\lfloor\frac n{jk}\right\rfloor$$

I simply need hints to start on this problem. I have verified that it's indeed true by checking a few examples and I can't seem to find a counter-example, so I'm almost sure that the problem is indeed correct.

However, I can't seem to think of a proper (and rigorous) proof of this. Can someone please help me by giving me a few hints?

I want to prove it myself, so I'd appreciate if people don't directly post the proof themselves. Also, please keep your hints as subtle as possible (there's no fun in proving something if the hint gives out most of the answer) :D

Hint: Let $n=kl+r$, where $l,r\in\mathbb{N}, 0\leqslant r<k$. Then $$\left\lfloor\frac n{k}\right\rfloor=l$$ Let $l=jp+q$, where $p,q\in\mathbb{N}, 0\leqslant q<j$ again and plug it in above.

Suppose $l=jp+q$, where $p,q\in\mathbb{N}, 0\leqslant q<j$. Then $$n=kjp+kq+r$$

where $0\leqslant kq+r<k(q+1)\leqslant kj$.

So $$ \left\lfloor\frac n{jk}\right\rfloor=p=\left\lfloor\dfrac{l}{j}\right\rfloor=\left\lfloor\dfrac{1}{j}\left\lfloor\dfrac{n}{k}\right\rfloor\right\rfloor $$