I would like to know if my thought on this problem are good. We have to study the convergence of the sequence $u_n = 5 - \frac{8}{5}n$. Intuitively it is clear that this sequence is not convergent in $\mathbb{R}$ and this is what I want to prove. Here is my idea : show that for an artificial limit point in $\mathbb{R}$, we can always go as far as we want to this point in some sense.
Suppose that $u_n$ converge to $l\in\mathbb{R}$. This means that
$$ \forall\epsilon>0, \exists N\in\mathbb{N} : n> N\implies\lvert u_n - l\rvert < \epsilon $$
Take $\epsilon = 1$, then there exists $N\in\mathbb{N}$ such that $n>N\implies\lvert u_n - l\rvert <1$. Now consider the distance between $u_n$ and $u_{n+2}$
$$ \lvert u_{n+2} - u_n\rvert = \left\lvert 5-\frac{8}{5}n - \left(5-\frac{8}{5}n -\frac{16}{5}\right)\right\rvert = \frac{16}{5} $$
It follows that
$$ \lvert u_{n+2} - l\rvert = \lvert u_{n+2} - u_{n} + u_n - l\rvert\geq\lvert\lvert u_{n+2} - u_{n}\rvert - \lvert u_n - l\rvert \rvert> \left\lvert\frac{16}{5} - 1\right\rvert>\frac{11}{5}> 1 $$
which contradicts the convergence of $u_n$ to $l$.
Is this proof correct and have you other idea to prove this which use others concepts ? Clearly my idea is from the notion of Cauchy sequence.
Thank you a lot for your help.