questions on mean-square convergence for a AR(P) example

Question

In the following example related to AR(P) process, I have two questions

I marked these two questions with two different colors.

Question 1) (I marked with yellow), why that sum is mean-square convergent? How to use the Cauchy criterion to get this point?

Question 2) (I marked with red), why this conclusion can be made?

Answer 1

1. Let $n > m$. By the triangle inequality, we have $$ \begin{align*} \left\| \sum_{j=0}^n \phi_1^j Z_{t-j} - \sum_{j=0}^m \phi_1^j Z_{t-j} \right\|_{L^2} &= \left\| \sum_{j=m+1}^n \phi_1^j Z_{t-j} \right\|_{L^2} \leq \sum_{j=m+1}^n |\phi_1^j| \|Z_{t-j}\|_{L^2} \end{align*}$$ By assumption, $(Z_t)_t$ is a white noise process; in particular $\|Z_{t-j}\|_{L^2} = \sigma^2>0$. Therefore, $$\left\| \sum_{j=0}^n \phi_1^j Z_{t-j} - \sum_{j=0}^m \phi_1^j Z_{t-j} \right\|_{L^2} \leq \sigma^2 \sum_{j=m+1}^n |\phi_1|^j.$$ As $|\phi_1|<1$, the latter is a Cauchy sequence (Hint: geometric series!), and this shows that $\left( \sum_{j=0}^n \phi_1^j Z_{t-j} \right)_{n \in \mathbb{N}}$ is a Cauchy sequence; hence convergent, i.e. $$\sum_{j=0}^n \phi_1^j Z_{t-j} \stackrel{L^2}{\to} \sum_{j=0}^{\infty} \phi_1^j Z_{t-j}. \tag{1}$$
2. It is shown that $$X_t - \sum_{j=0}^k \phi_1^j Z_{t-j} \stackrel{L^2}{\to} 0$$ On the other hand, we know from $(1)$ that $$\sum_{j=0}^{\infty} \phi_1^j Z_{t-j} - \sum_{j=0}^k \phi_1^j Z_{t-j} \stackrel{L^2}{\to} 0$$ By the uniqueness of the ($L^2$-)limit, this implies $$X_t = \sum_{j=0}^{\infty} \phi_1^j Z_{t-j}$$