is there a better way to prove it?

Question

$\{x^k\}$ converge super linaerly to $x^*$ meant $ \ \ \lim_{k\to\infty}\frac{||x^{k+1} - x^*||}{||x^k - x^*||^p}=r$ and $\ \ 0<p<2$ and $\ \ $$r$ is constant.

is It true that The value of below limit is equal to one؟

if $\{x^k\}$ converge super linaerly to $x^*$ then $$ \lim_{k\to\infty}\frac{||x^{k+1} - x^k||}{||x^k - x^*||}=1$$ ?

I wanted to prove it by definition, but I did not Should I prove through the definition? Or is there a better way?

Answer 1

If you split $x^{k+1}-x^{k}$ as $(x^{k+1}-x^{*})- (x^{k}-x^{*})$ you see that the limit is $|r-1|$, not 1. Perhaps you have stated the definition wrongly.