Finding an example where the sequences $(a_nc_n)$ and $(b_nc_n)$ converges to different values

Question

suppose we have sequences $(a_n)$, $b_n=a_{n+1}$, and $(c_n)$ where $c_n\neq 0 \forall n$. Furthermore $(c_n)$ has the property where $\lim_{n\to\infty}(c_n/c_{n+1})=1$.

One can show that $\lim_{n\to\infty}a_n = L \iff \lim_{n\to\infty}b_n = L$.

Let $x= Lc$ where $L$ is the limit of $(a_n)$ and $(b_n)$, and $c$ is the limit of $(c_n)$. Then

$\lim_{n\to\infty}a_nc_n = Lc =x \iff \lim_{n\to\infty}b_nc_n = Lc =x $

therefore if $(b_nc_n)$ converges, then $(a_nc_n)$ converges to the same limit

However, I'm not sure how it is possible to find an example where $(a_nc_n)$ and $(b_nc_n)$ converges to different values, since $a_n$ and $b_n$ have the same limit, then shouldn't $(a_nc_n)$ and $(b_nc_n)$ converge to the same value no matter what by the algebraic limit theorem?

Answer 1

Your argument is not correct since you assumed convergence of $(c_n)$ However, your guess is right. $(a_nc_n)$ and $(b_nc_n)$ cannot converge to different limits:

$b_nc_n=a_{n+1}c_n=\frac {a_{n+1}c_{n+1}} {c_{n+1}/c_n} $ which shows that $(a_nc_n)$ and $(b_nc_n)$ cannot have different limits. If either of these sequences converges so does the other, with the same limit.