I have a sequence $a_n\in\mathbb{N}$ which either converges to a value $a^{*}\in \mathbb{N}$ or diverges to $a^*=+\infty$. I also have a sequence $b_n\in \mathbb{R}^N$ for some $N \in \mathbb{R}$ which converges to $b^*$.
How can I show the following?
$$\lim_{n \rightarrow \infty}||b_n||_{a_n}=||b||_a$$
where $||\cdot||_p$ is a $p$-norm for $\mathbb{R}^N$ defined by some arbitrary inner product.
My progress so far:
I want to show that:
$$\forall \epsilon > 0, \exists N \in \mathbb{N} \text{ such that } n \geq N \implies \bigg|||b_n||_{a_n}-||b||_a\bigg|\leq \epsilon $$
I start with:
$$\bigg|||b_n||_{a_n}-||b||_a\bigg| = \bigg|||b_n||_{a_n}-||b_n||_{a}+||b_n||_{a}-||b||_a\bigg|$$ $$ \leq \bigg|||b_n||_{a_n}-||b_n||_{a}\bigg|+\bigg|||b_n||_a-||b||_a\bigg|$$
$$ \leq \bigg|||b_n||_{a_n}-||b_n||_{a}\bigg|+\epsilon/2$$
I think I can use the Cauchy-Schwarz inequality on the first term here, but this only works when $a_n\geq a_m$ for $n \leq m$. Am I missing something obvious here?
