A simple calculation showes that $$\sum_{f\in F(n+m,d)}A_f^*A_f=\sum_{g\in F(m,d)} A_g^*\left(\sum_{f\in F(n,d)}A_f^*A_f\right)A_g$$ Let $a_n:=\left\|\sum_{f\in F(n,d)}A_f^*A_f\right\|$. By use of the above equality we have $$0\leq a_{m+n}\leq a_ma_n$$ so $\lim_n a_n^{\frac 1n}$ exists.

A simple calculation shows that $$\sum_{f\in F(n+m,d)}A_f^*A_f=\sum_{g\in F(m,d)} A_g^*\left(\sum_{f\in F(n,d)}A_f^*A_f\right)A_g$$ Let $a_n:=\left\|\sum_{f\in F(n,d)}A_f^*A_f\right\|$. By use of the above equality we have $$0\leq a_{m+n}\leq a_ma_n$$ so $\lim_n a_n^{\frac 1n}$ exists.