(Solved): 1. (i). Let X=[0,1] Find a sequence (xn) in X such that xn1/2 and xn=xm for u ...

1. (i). Let $X=[0,1]$ Find a sequence $\left(x_n\right)$ in $X$ such that $x_n\to 1/2$ and $x_n\ne x_m$ for ull $n,m$ with $n\ne m$. (ii). Let $X=[0,1]\cup \{2\}$. Show that there is no sequence $\left(x_n\right){\hat{r}}^x$ with $x_n\to 2$ and $x_n\ne x_n$ wherever $n\ne m$. Give on example of a metric space $(x,d)$ such that for euch $x\in X$ and $r>0$, $B(x,r)\in \{X,\{x\}\}\text{. }$ Define a metric $p$ on $\mathbb{R}$ such that every function from $(\mathbb{R},p)$ into $(\mathbb{R},d)$, (where $d$ is usual metric on $\mathbb{R}$ ) is continuous.