I am self studying Banach Spaces of Analytic Function by Hoffman. In the Chapter 3 titled "Analytic and Harmonic Functions in the Unit Disc", the author defines the class $H^p$ for $1\le p \le \infty$ to be the class of analytic functions $f : \mathbb D \to \mathbb C$ for which the functions $f_r : \mathbb T \to \mathbb C$ defined by $f_r (e^{it}) = f(re^{it})$ for $re^{it} \in \mathbb D$ are uniformly bounded in the $L^p$ norm for $0 \le r < 1$, that is, there is some $M > 0$ such that for every $r\in [0,1)$, $\lVert f_r \rVert _p < M$.
The author defines a norm on $H^p$ in the following way: $\lVert f \rVert = \lim_{r\to 1} \lVert f_r \rVert _p$. It is not clear to me that the limit actually exists.
For $p= \infty$, I used the maximum modulus theorem for analytic functions to show that if $r_1 < r_2$ then $\lVert {f}_{r_1} \rVert _{\infty} \le \lVert {f}_{r_2} \rVert _ \infty$. Hence, the limit actually exists.
For $1 \le p < \infty$, I do not see how to prove it. Hints will be appreciated.