Exercise
13
[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:
\[ β \varepsilon {\gt}0, ~ β \overline nββ, ~ β nββ, nβ₯ \overline n \Rightarrow |a_ n|β€ \varepsilon |b_ n|~ . \]
Shown that these two clauses are equivalent.
Eventually in \(n\) we have that \(a_ n=0\iff b_ n=0\); having specified this, we have \(\lim _ n\frac{a_ n}{b_ n}=1\), 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.
Solution
1