If $a+ \varepsilon_n = (-a_n +b_n)\ge -b$ with $a, b\in\mathbb{R}^+, (\varepsilon_n)_n$ infinitesimal, then is true that $(-a_n+b_n)_n$ is bounded?

Question

Let $(a_n)_n, (b_n)_n$ be two sequences and $(\varepsilon_n)_n$ be an infinitesimal sequence. Consider the relation $$a+ \varepsilon_n = (-a_n +b_n)\ge -b, $$ for some positive real constants $a$ and $b$. During my calculus class, the lecturer said that the above inequality implies that the sequence $$ (-a_n +b_n)_n$$ is bounded.

It seems not correct to me since the above inequality gives $$(-a_n+b_n)\ge a+b+\varepsilon_n$$ and not $$(-a_n+b_n)\le a+b+\varepsilon_n.$$

What do you think about that? Could you please help me to understand if am I wrong or not?

Thank you in advance.

Answer 1

I think when the lecturer says $(\varepsilon_n)_n$ infinitesimal he/she means that $\epsilon_n$ are positive numbers tending to $0$.

$(\epsilon_n)$ is bounded. So we get $-a_n+b_n=a+\epsilon_n <a+M$ for some $M$. Also, $-a_n+b_n\geq -b$ so $-a_n+b_n$ is bounded.