Given that fulfills the following condition: there is a real number 0<<1 so for all-natural ≥2:
|$a_{n+1}−_|<c |(_^2−_{−1}^2)|$ where $|a_n|<2$
To prove : ($a_n$) is convergent
Now I was able to prove that ($a_n$) is convergent for the case where sum of consecutive terms of the sequence is more than or equal to one How should I go about solving the case where their sum will be less than one ??
This is false. Let $a_n=0$ for $n$ even and $a_n=1.5$ for $n$ odd. Let $\frac 2 3 <c<1$. Then $|a_{n+1}-a_n|=1.5 <c(1.5)^{2} <|a_n^{2}-a_{n-1}^{2}|$ for all $n$ but $(a_n)$ is not convergent.