How to prove a sequence is bounded For a sequence $\{x_n\}_{n\ge 1}$, let a sequence $\{y_n\}_{n\ge 1}$ be defined by the formula $$y_n = \dfrac{x_1 + x_2 + \ldots + x_n}n \ \forall n\in \Bbb{N}$$
Prove that if the sequence $\{x_n\}_{n\ge 1}$ is bounded, then so is $\{y_n\}_{n\ge 1}$. (Recall that a sequence $\{x_n\}_{n\ge 1}$ is bounded if there exists a positive real number $M$ such that $|x_n| < M$ $\forall n\in \Bbb{N}$ )

Because $\forall i \in \mathbb{N}, \ |x_i| < M$, then
$$\left|\dfrac{x_1 + x_2 + \ldots + x_n}n\right| \leqslant\frac{|x_1| + |x_2| + \ldots + |x_n|}{n} \leqslant\frac{M+M+\cdots+M}{n}=\frac{Mn}{n}=M$$
