calculus - Why does $lim_{ntoinfty} frac{a_{n+1}}{a_n} neq 1$ if $(a_{n+1})$ is a subsequence of $(a_{n})$?

According to the quotient rule, if $\lim_{n\to\infty} \frac{a_{n+1}}{a_n} \lt 1$, then the sequence $(a_n)$ must be monotonically decreasing (definitively) and converge to $0$. If the limit is greater than $1$, then the sequence must be monotonically increasing and diverge.

I can't understand why the limit would ever be different than $1$, since if $a_n$ converges to $l$, then any subsequence of said sequence is also convergent and converges to the same limit $l$. Consequently, both $(a_n)$ and $(a_{n+1})$ converge to the same limit, and here lies the crux of my question: why can $\lim_{n\to\infty} \frac{a_{n+1}}{a_n} \neq 1$ if both sequences converge to the same limit $l$?

Please anyone shed some light on this as I really can't wrap my head around it.