EDB — 02F

[02F]Prerequisites:[1Z7]↺↻.Let $a_n,b_n$ be real sequences (which can have variable signs, take value zero, and are not necessarily infinitesimal); let $X={ℝ}^{\mathbb{N}}$ the space of all sequences.

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

Show these results:

• for $a,b\in X,a=(a_n{)}_n,b=(b_n{)}_n$ consider the relation

  $$aRb⟺a_n=O(b_n)$$

  prove that $R$ is a preorder;

• define $x\asymp y⟺(xRy\wedge yRx)$ then $\asymp$ is an equivalence relation, $R$ is invariant for $\asymp$, and the projection $⪯$ is an order relation on $X/\asymp$ (hint: use the Prop. [1Z7]↺↻).

• Define (as usually done)

  $$\hat{a}\prec \hat{b}⟺(\hat{a}⪯\hat{b}\wedge \hat{a}\ne \hat{b})$$

  for $\hat{a},\hat{b}\in X/\asymp$, $(a_n{)}_n\in \hat{a},(b_n{)}_n\in \hat{b}$ representatives; assuming $b_n\ne 0$ (eventually in $n$), prove that

  $$\hat{a}\prec \hat{b}⟺0=\underset{n}{lim inf}\frac{a_n}{b_n}\le \underset{n}{lim sup}\frac{a_n}{b_n}<\infty .$$

The above discussion is related to Definition 3.2.3 (and following) in [ 3 ] .