Question about a transient response statement (Oppenheim-Schafer"s discrete-time signal processing book, 3rd ed)
It's true that Eq. $(129)$ just presents an upper bound, from which it can't directly be concluded that the transient response $y_t[n]$ decays to zero. However, stability requires that
$$\sum_{k=0}^{\infty}|h[k]|&lt;\infty\tag{1}$$
which can only be true if $\lim_{n\to\infty}h[n]=0$, i.e., if the impulse response decays to zero. If this is the case, then the transient response
$$y_t[n]=-\sum_{k=n+1}^{\infty}h[k]e^{j\omega (n-k)}$$
must also decay to zero for $n\to\infty$.