Show that $ \{(a_0,a_1,a_2, \ldots) \in \mathbb{R}^\infty \mid \sum\limits_{i = 0}^{\infty} |a_i|^p < \infty\}$ is a subspace of $\mathbb{R}^\infty$. Problem:

Let $\mathbb{R}^\infty = \{(a_0,a_1,a_2, \ldots) \mid a_i \in \mathbb{R}\}$, and for $p\in\{1,2,\ldots\}$, let $\mathscr{L}^p = \{(a_0,a_1,a_2, \ldots) \in \mathbb{R}^\infty \mid \sum\limits_{i = 0}^{\infty} |a_i|^p < \infty\}$.
a) Show that $\mathscr{L}^p$ is a subspace of $\mathbb{R}^\infty$.
b) Show that $\mathscr{L}^p$ is a proper subspace of $\mathscr{L}^{p+1}$

My work:
I first wrote the following proof and I thought that I was done with this problem, however you can easily realise that it is completely flawed.
First of all, clearly $\mathscr{L}^p \subset \mathbb{R}^\infty$ and $(0,0, \ldots) \in \mathscr{L}^p$ so in a) I just have to prove that $\mathscr{L}^p$ is closed under addition and scalar multiplication and, taking this into account, in b) I have to prove that $\mathscr{L}^p$ is a proper subset of $\mathscr{L}^{p+1}$ since both are vector spaces.
Consider any $(a_i)_{i\in \mathbb{N}}, (b_i)_{i\in \mathbb{N}} \in \mathscr{L}^p$, we first need to prove that $(a_i + b_i)_{i\in \mathbb{N}} \in \mathscr{L}^p$ or equivalently, $\sum\limits_{i = 0}^{\infty} |a_i + b_i|^p < \infty$ using the fact that $\sum\limits_{i = 0}^{\infty} |a_i|^p,\sum\limits_{i = 0}^{\infty} |b_i|^p < \infty$. Let's start by proving a non-trivial case but still simple, $p = 2$.
$$\sum\limits_{i = 0}^{\infty} |a_i + b_i|^2 \le \sum\limits_{i = 0}^{\infty} (|a_i| + |b_i|)^2 = \underbrace{\sum\limits_{i = 0}^{\infty} |a_i|^2 + \sum\limits_{i = 0}^{\infty} |b_i|^2 }_{< \infty} + 2\sum\limits_{i = 0}^{\infty} |a_i||b_i|$$
Now we know $\sum\limits_{i = 0}^{\infty} |a_i||b_i| \le \sum\limits_{i = 0}^{\infty} |a_i|\sum\limits_{j = 0}^{\infty} |b_j| < \infty$ so it is true for $p = 2$. Now let's assume it is true for all $p \le k$.
$$\sum\limits_{i = 0}^{\infty} |a_i + b_i|^{k+1} = \sum\limits_{i = 0}^{\infty} |a_i + b_i|^k|a_i + b_i| \le \sum\limits_{i = 0}^{\infty} |a_i + b_i|^k \sum\limits_{i = 0}^{\infty}|a_i + b_i| < \infty$$
By induction we conclude that the inequality holds for all $p \in \mathbb{N}$ so $\mathscr{L}^p$ is closed under addition. Furthermore, consider any $\lambda \in \mathbb{R}$
$$\sum\limits_{j = 0}^{\infty} |\lambda a_j|^p = \sum\limits_{j = 0}^{\infty} |\lambda|^p|a_j|^p = |\lambda|^p \sum\limits_{j = 0}^{\infty} |a_j|^p < \infty$$
So it is closed under scalar multiplication as well, and hence a vector space.
Let's consider any $(a_i)_{i\in \mathbb{N}} \in \mathscr{L}^p$ so we know $\sum\limits_{j = 0}^{\infty} |a_j|^p < \infty$.
$$\sum\limits_{j = 0}^{\infty} |a_j|^{p+1} = \sum\limits_{j = 0}^{\infty} |a_j| |a_j|^p \le \sum\limits_{j = 0}^{\infty} |a_j| \sum\limits_{j = 0}^{\infty} |a_j|^p < \infty$$
So if $(a_i)_{i\in \mathbb{N}} \in \mathscr{L}^p$ then $(a_i)_{i\in \mathbb{N}} \in \mathscr{L}^{p+1}$ and thus $\mathscr{L}^p \subset \mathscr{L}^{p+1}$.
Clearly $\mathscr{L}^p$ isn't the trivial subspace so we just have to prove $\mathscr{L}^p \neq \mathscr{L}^{p+1}$ that can be done by showing that $(1/\sqrt[p]{n})_{n \in \mathbb{N}} \in \mathscr{L}^{p+1}$ but does't belong to $\mathscr{L}^p$.
So after writing it I read it and found out that all the proof was wrong because I was reckless and made the silly and nonsensical assumption $(a_i)_{i\in \mathbb{N}} \in \mathscr{L}^p \implies \sum\limits_{j = 0}^{\infty} |a_j| < \infty$ that is completely wrong since the $\sum\limits_{j = 0}^{\infty} |a_j|^p$ converges because all but finitely many elements $a_i$ are less than $1$ so intuitively,
$$\sum\limits_{j = 0}^{\infty} |a_j|^{p+1} \le \sum\limits_{j = 0}^{\infty} |a_j|^{p} < \infty$$
And proving this rigorously would fix part b) of the problem, however I can't figure out how to do this nor how to prove that $\mathscr{L}^p$ is closed under addition without making that silly and wrong assumption.
I would hugely appreciate any hints.
Thanks in advance.
 A: a) For the closeness under addition, I don't think your proof by induction works. You want to show
$$\forall(\mathbf a,\mathbf b),\qquad \mathbf a,\mathbf b\in\cal L^k\implies\mathbf a+\mathbf b\in\mathcal L^k.$$
I don't see how an induction over $k$ could help. The fact that $\mathbf a,\mathbf b\in\mathcal L^{k+1}$ does not imply either $\mathbf a+\mathbf b\in\mathcal L^k$ or $\mathbf a+\mathbf b\in\mathcal L^1$. Think of $k=1$ and $\mathbf a=(1,0,\frac13,0,\frac15,\ldots)$ and $\mathbf b=(0,\frac12,0,\frac14,\ldots)$.
However you can use this consequence of Jensen's inequality:
let $p\ge1$, and let $\mathbf a,\mathbf b\in\mathbb R^\infty$. Then for every $n\ge1$
$$\sum_{i=1}^n|a_i+b_i|^p\le2^{p-1}\left(\sum_{i=1}^n|a_i|^p+\sum_{i=1}^n|b_i|^p\right)\!.$$
Next pass to the limit $n\to\infty$.

b) For $p\in[0,\infty)$, write $$\|\mathbf a\|_p^p:=\sum_{i=1}^\infty|a_i|^p.$$
If $\|\mathbf a\|_p<\infty$, then
$$\sum_{i=1}^\infty{\left(\frac{|a_i|}{\|\mathbf a\|_p}\right)}^p=1.$$
In particular all terms in this series are $\le1$, and so
$${\left(\frac{|a_i|}{\|\mathbf a\|_p}\right)}^{p+1}\le{\left(\frac{|a_i|}{\|\mathbf a\|_p}\right)}^p\le1.$$
You should be able to conclude from here.
