Please help me to understand the basic concepts of the following sequence (apologies in advance for asking simple things probably, but it is heavy for me yet. With your blessings, I can learn more and understand things.):
The sequence $\left\{ f (k) \right\}$ of functions in $C[0,1]$, i.e, the vector space of all complex-valued or real-valued continuous functions on $[0, 1]$ is given by
\begin{align*} & f_k(x) = 0, & \ 0 \leq x \leq \frac{1}{k} \\ & f_k(x) = 2\left(k^{\frac{3}{2}}x - k^{\frac{1}{2}} \right), & \ \frac{1}{k} \leq x \leq \frac{3}{2k} \\ & f_k(x) = 2\left(-k^{\frac{3}{2}}x + 2k^{\frac{1}{2}} \right), & \ \frac{3}{2k} \leq x \leq \frac{2}{k} \\ & f_k(x) = 0, & \ \frac{2}{k} \leq x \leq 1 .\\ \end{align*}
Now, please help me to understand how and why the following two are true, namely
(i) $\lVert f_k(x) - f_j(x)\rVert_1 \rightarrow 0$ as $k, \ j \rightarrow \infty$. Why and how?
(ii) for each $k \geq 2$, there is some $J > k$ for which $\lVert f_k(x) - f_j(x)\rVert_{\infty} > k^{\frac{1}{2}}$ for all $j > J$. Hmm, why and how?
Many thanks in advance for your help