Problem 5, Chapter 3. Rudin's functional analysis, point (a) and (b)

Need an hint on this exercise, only first two points for now.

For $0 < p < \infty$, let $l^p$ be the sapce of all functions $x$ (real or complex, as the case may be) on the positive integers, such that $$ \sum_{n=1}^{\infty} |x(n)|^p < \infty $$ For $0 \leq p < \infty$. Define $\left\lVert x \right\rVert = \left\{ \sum |x(n)|^p \right\}^{1/p}$, and define $\left\lVert x \right\rVert_{\infty} = \sup_n |x(n)|$.
(a) Assume $1 \leq p < \infty$. Prove that $\left\lVert x \right\rVert_p$ and $\left\rVert x \right\rVert_{\infty}$ make $l^p$ and $l^{\infty}$ into Banach Spaces. If $p^{-1} + q^{-1} = 1$, prove that $(l^p)^* = l^q$ in the following sense: There is a one-to-one correspondence $\Lambda \leftrightarrow y$ between $(l^p)^*$ and $l^q$, given by $$ \Lambda x = \sum x(n)y(n) \;\;\; (x \in l^p) $$
(b) Assume $1 < p < \infty$ and prove that $l^p$ contains sequences that converge weakly but not strongly.

I think (a) is trivial. Since $L^p(X,\mu)$ is a banach space taking as $\mu$ the counting measure we obtain the result. Likewise Riesz representation with the counting measure $\mu$ gives me the representation of the dual space.
For (b) I am not sure, my thought was to construct an example of sequence $$ x_m(n) = \begin{cases} 1 & m \leq n \\ 0 & \text{otherwise} \end{cases} $$

I ended up with nothing, because I don't know how to show eventually this converges weakly. (I know that this is probably a standard exercise, my apologies if the question is trivial).

Solution 1:

Take $x_m(n)=1$ if $n=m$ and $0$ otherwise. This sequence converges weakly to $0$ becasue $\sum x_m(n)y_n =y_m \to 0$ as $ m \to \infty$ for any $y=(y_n)$ in the dual of $\ell^{p}$ (which is $\ell^{q})$. Of course, this sequence does not converge strongly.