I'm having a lot of trouble to do that extra practicing exercise. Hope some of you could help me to do and understand it!
So, let $\textsf{C}([a, b]) := \{ f:[a,b]\to\Bbb{R} :\, f \textrm{ is continuous}\}$.
We define a function $d: \textsf{C}([a, b])\times\textsf{C}([a, b])\to\Bbb{R}$ as $$d(f,g) := \left(\int(f-g)^2\right)^{1/2} \quad \textrm{ for all } f,g\in\textsf{C}([a, b])$$
since $(f − g)^2$ is continuous, so it's integrable when $f$ and $g$ are too.
Knowing this, I have to show the following properties.
$\textbf{1.}$ Show that $d(f,g) ≥ 0$ no matter what $f,g\in\textsf{C}([a, b])$.
$\textbf{2.}$ Show that if $f,g\in\textsf{C}([a, b])$ then $d(f,g) = 0$ if and only if $f = g$.
$\textbf{3.}$ Show that $d(f, g) ≤ d(f, h) + d(g, h)$ knowing that $f,g,h\in\textsf{C}([a, b])$.
If $f,g \in C[a,b]$, here are two useful inequalities: $$\int (f-g) dx \leq M (b-a)$$ and $$\int (f-g) dx \geq m (b-a)$$ where $M$ denotes $sup(f-g)$ and $m$ denotes $inf(f-g)$.
For 1, apply the estimate to $(f-g)^2$.
For 2, the hard direction can be shown by using the second fundamental theorem of calculus.