(a) Let $\delta>0$ and $x_{0} \in \mathbb{R}$. Let $\left\{\phi_{n}(x)\right\}_{n=0}^{\infty}$ be a sequence of (real) functions that are nonzero for all $x$ with $0<\left|x-x_{0}\right|<\delta$, and let $\left\{a_{n}\right\}_{n=0}^{\infty}$ be a sequence of nonzero real numbers. For every $N=0,1,2, \ldots$, the function $f(x)$ satisfies

$$f(x)-\sum_{n=0}^{N} a_{n} \phi_{n}(x)=o\left(\phi_{N}(x)\right), \quad \text { as } \quad x \rightarrow x_{0}$$

(i) Show that $\phi_{n+1}(x)=o\left(\phi_{n}(x)\right)$, for all $n=0,1,2, \ldots$; i.e., $\left\{\phi_{n}(x)\right\}_{n=0}^{\infty}$ is an asymptotic sequence.

(ii) Show that for any $N=0,1,2, \ldots$, the functions $\phi_{0}(x), \phi_{1}(x), \ldots, \phi_{N}(x)$ are linearly independent on their domain of definition.

(b) Let

$$I(\varepsilon)=\int_{0}^{\infty}(1+\varepsilon t)^{-2} e^{-(1+\varepsilon) t} d t, \quad \text { for } \varepsilon>0$$

(i) Find an asymptotic expansion (not necessarily a power series) of $I(\varepsilon)$, as $\varepsilon \rightarrow 0^{+}$.

(ii) Find the first four terms of the expansion of $I(\varepsilon)$ into an asymptotic power series of $\varepsilon$, that is, with error $o\left(\varepsilon^{3}\right)$ as $\varepsilon \rightarrow 0^{+}$.