I'm aware of the conditions of a CDF. to be a cumulative distribution: it is always nonnegative, when x→−∞ x→−∞ it tends to 0, when x→+∞ it tends to 1, and it is right-continuous .
The CDF is proportional to $x$ if it has the form $F(x)=cx$, for some constant $c$. You need that, for $x<0$, $F(x)=0$, and for $x>1$, $F(x)=1$.
What constant $c$ makes it such that $F(0)=0$, $F(1)=1$?
$F(1)=c\cdot1=1$, so we need $c$ to be equal to 1.