Given a real sequence $(a_n)_n$ converging to a finite value $a$, a property of the Cesàro mean, defined as the arithmetic mean
$$ b_n=\frac{a_1+\ldots+a_n}{n}, $$
is
$$ \lim_{n\to\infty}b_n=a,\tag1 $$
so that, supposing $a_n\neq0$ $\forall\,n$ and $a\neq0$, we can also deduce
$$ \lim_{n\to\infty}\frac{b_n}{a_n}=1.\tag2 $$
Is result $(2)$ also valid for $a=0$?
And are results $(1)$ and/or $(2)$ also valid for $a=+\infty$?