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

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?

• It is not given that $(c_n)$ is convergent. Apr 24, 2021 at 7:24
• @KaviRamaMurthy How can one prove that $(a_kc_k)$ converges then? If we consider $a_k(c_k/c_{k+1})=L\cdot 1$ by the algebraic limit theorem, then $a_kc_k=Lc_{k+1}$. However, this doesn't yield anything useful. Apr 24, 2021 at 7:29
• Are you sure it is $(c_n/c_{n+1}) \to 1$ and not $|c_n/c_{n+1}| \to 1$ ? Apr 24, 2021 at 7:32
• @nicomezi yes, its given that $(c_n/c_{n+1})\to 1$ and not $|c_n/c_{n+1}|\to 1$ Apr 24, 2021 at 7:34

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.

• I see, this also implies that it is not possible for $(a_nc_n)$ to converge while $(b_nc_n)$ diverges, correct? Apr 24, 2021 at 7:42
• @bruhmoment28 Yes, if one of then converges the other one also converges and the limits are same. Apr 24, 2021 at 7:44