MHB Limits of x* when n and m Approach Infinity

[ozkan12]

if $\lim_{{n}\to{\infty}} {y}_{m+1}={x}^{*}$, Can we say that $\lim_{{n}\to{\infty}} {y}_{n-1}={x}^{*}$...Please pay attention, I say m and n...

 

[Ackbach]

if $\lim_{{n}\to{\infty}} {y}_{m+1}={x}^{*}$, Can we say that $\lim_{{n}\to{\infty}} {y}_{n-1}={x}^{*}$...Please pay attention, I say m and n...

Did you mean to write

If
$$\lim_{m\to\infty}y_{m+1}=x^*,$$
can we say that
$$\lim_{n\to\infty}y_{n-1}=x^*?$$

Because, if so, the answer is yes, if $n$ and $m$ are just independent integers going to infinity. The variable used in the limit is just like a "dummy variable". You can also show that if $n$ is an arbitrary integer, and
$$\lim_{n\to\infty}y_n=x^*,$$
then any subsequence must also converge:
$$\lim_{m\to\infty}y_{n_m}=x^*.$$

 

[ozkan12]

Dear Ackbach

How if $\lim_{{m}\to{\infty}} {y}_{m+1}={x}^{*}$ we can say that $\lim_{{n}\to{\infty}} {y}_{n}={x}^{*}$ ? I didnt understand...Please can you explain...Thank you for your attention