Let \(\lambda\) be Lebesgue measure in \((0, \infty)\) and \(p, q \geqslant 1\) arbitrary.

(i) Show that \(u_{n}(x):=n^{\alpha}(x+n)^{-\beta}(\alpha \in \mathbb{R}, \beta>1)\) holds for every \(n \in \mathbb{N}\) in \(\mathcal{L}^{p}(\lambda)\).

(ii) Show that \(v_{n}(x):=n^{\gamma} e^{-n x}(\gamma \in \mathbb{R})\) holds for every \(n \in \mathbb{N}\) in \(\mathcal{L}^{q}(\lambda)\).