real analysis - Proof of Raabe's Test for divergence

Question: Prove that if $u_n>0$ and $\lim_\limits{n\to\infty}n\left(\frac{u_n}{u_{n+1}}-1\right)=l$ , then $\sum_{n=1}^{\infty} u_n$ diverges if $l<1$.

My Attempt: We choose a positive number $\epsilon$ such that $l+\epsilon<1$.

Since $\lim_\limits{n\to\infty}n\left(\frac{u_n}{u_{n+1}}-1\right)=l$, then for any $\epsilon>0$ , there exists a natural number $m$ such that

$$\left| \,\ n\left(\frac{u_n}{u_{n+1}}-1\right)-l\,\ \right|< \epsilon \,\ \forall \,\ n\ge m$$

Therefore, we have
$$l-\epsilonNow say

$$l+\epsilon=r$$ Hence $$r<1$$ So we can say that $$n\left(\frac{u_n}{u_{n+1}}-1\right)

Any suggestions on how to proceed from here?

P.S. I have read other questions here on MathsSE regarding this proof specifically but I am not satisfied with these answers as one uses big O notation and another has a different form of Raabe's test. I don't want answers with big O notation. Proofs with other methods are welcome but the ones following from where I have ended are most preferred.

Answer

Your first steps are all correct.
You have obtained the inequality

$$\frac{u_n}{u_{n+1}} < \frac{r}{n} + 1$$
for $n\geq m$
but have realized that, in this form, it is rather hard to do anything with it.

Here is the classical argument as to how to proceed.

Rewrite this as

$$n \frac{u_n}{u_{n+1}} -(n+1) < r-1 \leq 0.$$
Then observe that
$$nu_n \leq (n+1)u_{n+1}$$
for all $n$ greater than $m$. Thus we have an increasing
sequence $\{nu_n\}$ and, in particular there is some positive constant $c$
for which $nu_n\geq c$.

Thus for $n\geq m$ we have

$$u_n \geq \frac{c}{n}$$
and a comparison with the harmonic series establishes
divergence as you desired.

There is a more general version of Rabbe's test known as
Kummer's test with very much the same proof.
For that you assume there is a sequence of positive numbers
$D_n$ and you compute the limit

$$L =\lim_{n\to\infty} \left[ D_n\frac{u_n}{u_{n+1}} -D_{n+1} \right]$$

Raabe's test is just Kummer's test with $D_n=n$.

The divergence part of Kummer's test (and hence also Raabe's test) doesn't actually require limits. You simply need that

$$D_n\frac{u_n}{u_{n+1}} -D_{n+1} \leq 0$$
for all sufficiently large $n$ and the divergence of $\sum_{n=1}^\infty 1/D_n$ and you can conclude divergence.