Let $p$ be a fix number such that $1\leq p<\infty$ and let $(\textbf{a}_n)_{n\in\mathbb{N}}$ be a sequence in $l_p$ where $\textbf{a}_n=(a^{(k)}_n)_{k\in\mathbb{N}}$ for all $n\in\mathbb{N}$. Assume that $\textbf{a}_n$ converges componentwise to $\textbf{a}=(a^{(k)})_{k\in\mathbb{N}}$, i.e., $\lim_{n\rightarrow\infty}a^{(k)}_n=a^{(k)}$ for any $k\in\mathbb{N}$. Then prove the following statements:
(a) $\textbf{a}$ is not necessarily in $l_p$.
(b) $\textbf{a}_n$ need not converge to $\textbf{a}$ in $l_p$ even though $\textbf{a}$ is assumed to be in $l_p.$
(c) Suppose that, in addition to the above, there is a sequence $\textbf{b}=(b^{(k)})_{k\in\mathbb{N}}$ in $l_p$ such that $|a^{(k)}_n|\leq b^{(k)}$ for all $n,k\in\mathbb{N}$. Prove that $\textbf{a}\in l_p$ and the sequence $\textbf{a}_n$ converges to $\textbf{a}$ in $l_p.$
Here is my attempt for (a), (b), and first part of (c). Can someone give me comments and advice to improve my argument more rigorously, if any.
(a) Let $\textbf{a}_n=(1,\frac{1}{n},...,\frac{1}{n},0,...)\in l_p$ (where $\frac{1}{n}$ shows up $n$ terms) and $\textbf{a}=(1,0,0,...)$. We see that $\|\textbf{a}_n-\textbf{a}\|_p=\left(\frac{1}{n^p}+\cdots+\frac{1}{n^p}\right)^{1/p}=\left(\frac{n}{n^p}\right)^{1/p}\rightarrow\infty$ as $n\rightarrow\infty.$ But $\|\textbf{a}\|_p=\left(\sum_{n=1}^{\infty}1\right)^{1/p}\rightarrow \infty$ which means that $\textbf{a}\notin l_p.$
(b) Take $\textbf{a}_n=(1,2,3,…,n,0,0,…)\in l_p$ and $\textbf{a}=(1,\frac{1}{2},\frac{1}{3},….,\frac{1}{n},…)\in l_p$. However, $\|\textbf{a}_n-\textbf{a}\|^p_p=(2-\frac{1}{2})^p+(3-\frac{1}{3})^p+\cdots+(n-\frac{1}{n})^p+\frac{1}{(n+1)^p}+\cdots>1$ which implies $\textbf{a}_n\not\rightarrow\textbf{a}$ in $l_p$.
(c) Since $\textbf{a}_n$ converges componentwise to $\textbf{a}$, we then have $|a^{(k)}_n|\leq b^{(k)}$ by taking $n\rightarrow\infty$ in the given bound for $|a^{(k)}_n|$. Notice that $\|\textbf{a}\|^p_p=\sum_{n=1}^{\infty}|a^{(k)}_n|^p\leq \sum_{n=1}^{\infty}(b^{(k)})^p<\infty$. This gives us $\textbf{a}\in l_p.$
The second part (showing that $\textbf{a}_n\rightarrow\textbf{a}$ in $l_p$) has not been yet proved! Any ideas and suggestions are all welcome.