Show the sequence converges

Question

We have a sequence $(x_{n})_{n\geq1}$ with all terms strictly positive and we know the sequence is structly decreasing and has the limit 0. Prove that the sequence $y_{n}= x_{1} + x_{2}/2 + x_{3}/3 +...+x_{n}/n$ converges. I found that $(y_{n})_{n\geq1}$ is strictly increasing.I tried some inequalities on every term of the sum and I got $x_{1}H_{n}>y_{n}>x_{n}H_{n}$ which does not help us. Then I tried CBS inequality and I got that$y_{n}^2<(x_{1}^2+x_{2}^2+...+x_{n}^2)(1+1/2^2+1/3^2+...+1/n^2)$. I tried again some inequalities on $(x_{1}^2+x_{2}^2+...+x_{n}^2)$ but nothing. Maybe we can suppose the $(y_{n})_{n\geq1}$ diverges to get a contradiction using the negation of the definition of Cauchy sequence?

Answer 1

Unfortunately, $\{y_n\}$ may not converge. Consider $x_n = \frac{1}{\ln n}$ for $n \geq 3$.