Can we relax the condition of Cesàro's Lemma?

Question

Cesàro's Lemma: Suppose that ($b_n$) is a monotonic increasing sequence of strictly positive real numbers such that $b_n \to \infty$. Let $x_n \to x < \infty$, then: $$\frac{1}{b_n}\sum_{k=1}^{n}{(b_k-b_{k-1})x_k}\to x \quad (n \to \infty).$$

Proof is here: proof.

However, I think that the condition of $b_n$, which is a positive monotonic increasing sequence, is too strict, all the proof need is $b_n \to +\infty$.

So is the result still right if I change the condition to $\lim b_n = +\infty$, that is,

$\forall G>0, \exists N > 0$, such that $b_n > G, \forall n > N.$

Answer 1

Merge the two sequences $(a_n)=(n)$ and $(c_n)=(2^n)$ as one being the even and the other being the odd terms of a new sequence $(b_n)_n$. Set $x_n$ to be a positive constant for all $n$. What result do you obtain?

In the proof, the monotonicity is assumed in the first inequality where the liminf is involved as you need each term of the sum to be positive.