We know that
$$|l(x+u)-l(x)|<1 \text{ for } x\geq y>0 \text{ and } u\in[0,1]$$
Why does:
$$|l(y+u)|<1+|l(y)|,x \in (y+1,y+2)$$
imply that
$$|l(x)| \leq 1 + |l(y+1)| \leq 2+|l(y)|$$
We know that
$$|l(x+u)-l(x)|<1 \text{ for } x\geq y>0 \text{ and } u\in[0,1]$$
Why does:
$$|l(y+u)|<1+|l(y)|,x \in (y+1,y+2)$$
imply that
$$|l(x)| \leq 1 + |l(y+1)| \leq 2+|l(y)|$$
This is true for any inequality of the form $|a-b|<1$. Since $|a-b|=|b-a|$ we can say both that $|a|<1+|b|$ and $|b|<1+|a|$. Then combining the two statements give that $|b|<1+|a|<1+(1+|b|)=2+|b|$