In my lecture notes, the professor says:
Let $\{f_n\}_{n=1}^{\infty}$ be a Cauchy sequence of continuous functions $f:[t_0,t_1]\to \mathbb R^n$. For each $t\in[t_0,t_1]$, $\{f_n(t)\}_{n=1}^{\infty}$ is a Cauchy sequence of vectors in $\mathbb R^n$.
Question : I'd like to know why is this the case?
Note that we define $\{v_i\}_{i=0}^{\infty}$ to be a Cauchy sequence if and only if
$$ \forall\varepsilon>0\,\,\exists N\in \mathbb N\text{ such that }\forall m\ge N,\,\,\|v_m-v_N\|<\varepsilon $$
Please take the infinity norm:
$$ \|f\|_\infty=\max_{t\in[t_0,t_1]}{\|f(t)\|_2} $$
When we talk about a Cauchy sequence, we need to equip a norm on the space.
You are saying about the space $X=\{f| f:[t_0,t_1]\rightarrow R^n\}$
The important thing is what is the norm on $X$.
In this case, since the norm is not specified, your professor maybe intend to say about the Cauchy sequence pointwisely.
Just my opinion. Thank you.