EDB — 0DJ

[0DJ]Let $a_n,b_n$ be real sequences (which can have variable signs, take value zero, and are not necessarily infinitesimal).

Recall that the notation $a_n=o(b_n)$ means:

Shown that these two clauses are equivalent.

• Eventually in $n$ we have that $a_n=0⟺b_n=0$; having specified this, we have $\underset{n}{lim}\frac{a_n}{b_n}=1$, where it is decided that $0/0=1$ (in particular $a_n,b_n$ eventually have the same sign, when they are not both null);

• we have that $a_n=b_n+o(b_n)$.

The second condition appears in Definition 3.2.7 in [ 3 ] where it is indicated by the notation $a_n\sim b_n$.

Deduct that $a_n\sim b_n$ is an equivalence relation.