Let $(x_n)$ be a sequence of real numbers converging to $x$.
Define sequences $y_n:=\max\{x_1,x_2,\ldots,x_n\}$ and $z_n:=\min\{x_1,x_2,\ldots,x_n\}$
Now do the sequences $(y_n),(z_n)$ converge? If $(x_n)$ is monotonic , then both these sequences are convergent. What can we say in general case? Please help me with this. Thanks!
$(y_n)$ is monotically increasing. It is also bounded: There exists $N$ such that $\lvert x_n - x \rvert < 1$ for $n \ge N$. Thus $x_n < x + 1$ for $n \ge N$ and thefore $y_n \le \max(y_N,x+1)$ for all $n$. Thus $(y_n)$ converges to some $y \in \mathbb R$. Similarly $(z_n)$ converges to some $z \in \mathbb R$.
However, $z < y$ unless $(x_n)$ is constant: If $(x_n)$ is not constant, then $x_N \ne x_M$ for suitable $N, M$. Let $R = \max(N,M)$. Then $z_R < y_R$ and thus $$z \le z_R < y_R \le y .$$