I have to deal with a problem for Real Analysis of Royden and Fitzpatrick. It is Problem 12 from Section 7.2.
For $1 \le p < \infty$ and a sequence $a = (a_1, a_2,\ldots) \in \ell^p$, define $T_a$ to be the function on the interval $[1,\infty)$ that takes the value $a_k$ on $[k,k+1)$, for $k=1,2,\ldots$. Show that $T_a \in L^p[1,\infty)$ and $\|a\|_p = \|T_a\|_p$.
This means I have to show $T_a \in L^p[1,\infty)$ and $$\int_{[1,\infty)} \Big|\sum_{k=1}^{\infty} a_k \chi_{[k,k+1)}\Big|^p = \sum_{k=1}^{\infty} |a_k|^p,$$ where $T_a = \sum\limits_{k=1}^{\infty} a_k \chi_{[k,k+1)}$ on $[1,\infty)$.
Here is my attempt.
I think I can show it by using Minkowski's inequality.
$$\Big\| \sum\limits_{k=1}^{n} a_k \chi_{[k,k+1)} \Big\|^p_p \le \sum\limits_{k=1}^{n} \Big\| a_k \chi_{[k,k+1)} \Big\|^p_p \le \sum_{k=1}^{\infty} |a_k|^p.$$
Let $n \to \infty$, by Monotone Convergence Theorem, we obtain $\|T_a\|_p \le \|a\|_p < \infty$, which implies $T_a \in L^p$.
But then, I got stuck.
Does anyone have any advice? I really appreciate it.