# Convergence of Norm Sequences

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:

$$\forall \epsilon > 0, \exists N \in \mathbb{N} \text{ such that } n \geq N \implies \bigg|||b_n||_{a_n}-||b||_a\bigg|\leq \epsilon$$

$$\bigg|||b_n||_{a_n}-||b||_a\bigg| = \bigg|||b_n||_{a_n}-||b_n||_{a}+||b_n||_{a}-||b||_a\bigg|$$ $$\leq \bigg|||b_n||_{a_n}-||b_n||_{a}\bigg|+\bigg|||b_n||_a-||b||_a\bigg|$$
$$\leq \bigg|||b_n||_{a_n}-||b_n||_{a}\bigg|+\epsilon/2$$
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?