The boundedness of Lusztig's $a$-function for Coxeter groups of finite rank

We prove that there exists a bound $N_L^{\mathrm{\prime }}(W)$ for a positively weighted Coxeter group $(W,S,L)$ of finite rank. In particular, Lusztig’s $\mathit{a}$-function of $(W,S,L)$ is bounded.