Let $x_{n}(n=1,2, \ldots)$ be real numbers.

What does it mean to say that the sequence $\left(x_{n}\right)_{n=1}^{\infty}$ converges?

What does it mean to say that the series $\sum_{n=1}^{\infty} x_{n}$ converges?

Show that if $\sum_{n=1}^{\infty} x_{n}$ is convergent, then $x_{n} \rightarrow 0$. Show that the converse can be false.

Sequences of positive real numbers $x_{n}, y_{n}(n \geqslant 1)$ are given, such that the inequality

$$y_{n+1} \leqslant y_{n}-\frac{1}{2} \min \left(x_{n}, y_{n}\right)$$

holds for all $n \geqslant 1$. Show that, if $\sum_{n=1}^{\infty} x_{n}$ diverges, then $y_{n} \rightarrow 0$.