i have a question about the following assertion:

let $n,j,u $ positive integer satisfying

$ n \geq 5,$ $ 1\leq j \leq n-1$,$ \; n+1 \leq u \leq n+j$

let $ d[n]:=\operatorname{lcm}[1,2,..,n]$ thus $u$ divide $d[n] \cdot C_{n+j}^n$

I think I have found a proof using valuation p-adic of prime number appearing in $u$ but I would like have another proof... thanks for help