# Convergent sequences of points

Let $$(a_{n)n \in N}$$ and $$(b_{n)n \in N}$$ be two convergent sequences of points in $$\mathbb{R}^{d}$$. Prove if $$a_n \rightarrow a$$ and $$b_n \rightarrow b$$, then $$||a_n - b_n|| \rightarrow ||a-b||$$ as $$n \rightarrow \infty$$

I know that if they $$\lim_{n \to \infty} a_n \leq \lim_{n \to \infty} b_n$$ for only sufficiently large values of n using the squeeze theorem. Would the same apply here or is there something else to be considered?

If $$a_n \to a$$ and $$b_n \to b$$, it is clear that $$a_n - b_n \to a-b$$. To see this, consider the inequality $$\|a_n - b_n - a+b\| \le \|a_n - a\| + \|b_n - b\|$$ for all $$n\in \mathbb N$$. Hence, your question boils down to showing that if $$c_n \to c$$, then $$\|c_n\| \to \|c\|$$. The triangle inequality comes to our rescue once again. We have $$\|c_n\| \le \|c_n - c\| + \|c\| \tag{1}$$ and $$\|c\| \le \|c_n - c\| + \|c_n\| \tag{2}$$ Combining $$(1)$$ and $$(2)$$, we get $$|\|c_n\| - \|c\|| \le \|c_n - c\|$$ so that when $$\|c_n - c\| \to 0$$, we also have $$\|c_n\| \to \|c\|$$. Apply this result to $$c_n = a_n - b_n$$ to get the desired conclusion.