We have the sequence with strictly positive real numbers $(x_{n})_{n\geq1}$. We know that there is a $k>0$ such that $x_{n}^{2}\leq k(x_{n}-x_{n+1})$ for every $n\geq1$. Prove that the sequence $(y_{n})_{n\geq1}$, $$y_{n}=x_{1}+\frac{x_{2}}{2}+\dots+\frac{x_{n}}{n}$$ is convergent.
I first showed that the limit of $x_{n}$ is $0$, because from the inequality we can find that the sequence is strictly decreasing and by using the fact that all terms of $x_{n}$ are strictly positive we can say $x_{n}$ is convergent. Taking the limit in the inequality we get that the limit is $0$. Now I am stuck at proving $y_{n}$ is convergent. I tried to use the squeeze theorem using inequalities with the harmonic series but it does not work.
From the given inequality we get that the series $\sum_{n=1}^{\infty}x_n^2$ converges. By Cauchy-Schwartz $$\sum_{n=1}^\infty \frac{x_n}{n} \leq \sqrt{\sum_{n=1}^\infty \frac{1}{n^2}}\sqrt{\sum_{n=1}^\infty x_n^2}$$ and since both series on the RHS converge the series $\sum_{n=1}^\infty \frac{x_n}{n}$ converges.