Let $F:[0,1] \to \mathbb{R}^n$ be a continuous function such that $F=(f_1,...,f_n)$ and let:$$\int_0^1 F(t) dt := \left( \int_0^1 f_1(t) dt,...,\int_0^1 f_n(t) dt\right)$$And:$$\lVert F \rVert_{\infty} := \sup_{t \in [0,1]} \lVert F(t) \rVert$$
(Where $\lVert F(t) \rVert = \left(\sum_{i=1}^n (f_i(t))^2 \right)^{\frac{1}{2}}$ )
Show that:
$$\left \lVert \int_0^1 F(t) dt\right \rVert \leq \lVert F \rVert_{\infty}$$
My attempt
\begin{equation*} \begin{split} \left \lVert \int_0^1 F(t) dt\right \rVert &= \left( \sum_{i=1}^n \left( \int_0^1 f_i(t) dt \right)^2\right)^{\frac{1}{2}} \\& \leq \sum_{i=1}^n \left| \int_0^1 f_i(t) dt \right| \\& \leq \sum_{i=1}^n \int_0^1 |f_i(t)| dt \\&= \sum_{i=1}^n \lVert f_i \lVert_1 \end{split}\end{equation*}
And since $F$ is continuous each function $f_i$ is continuous on $[0,1]$ so by the Hölder inequality: $\lVert f_i \lVert_1 = \lVert f_i \cdot 1 \rVert_1 \leq \lVert 1 \rVert_1 \cdot \lVert f_i \rVert_{\infty}$, where $\lVert f_i \rVert_{\infty} = \sup_{t \in [0,1]} |f_i(t)|$. So we have:
\begin{equation*} \begin{split} \left \lVert \int_0^1 F(t) dt\right \rVert & \leq \sum_{i=1}^n \lVert f_i \lVert_1 \\& \leq \sum_{i=1}^n \lVert 1 \rVert_1 \cdot \lVert f_i \rVert_{\infty} \\& = \sum_{i=1}^n\int_0^1 1 dt \cdot \lVert f_i \rVert_{\infty} \\&= \sum_{i=1}^n \lVert f_i \rVert_{\infty} \end{split}\end{equation*}
This was as far as I got. I don't know how to make $\lVert F \rVert_{\infty}$ appear. I would appreciate some help