I have previously shown that if $d_1$ and $d_2$ are two metrics in the set $M$ then $D(x,y)=\max\{d_1(x,y),d_2(x,y)\}$ is also a metric in $M$, but I am not sure how to progress with the following problems:
i)
Let $a\in M$ and $r>0$. Let $K_1(a,r),K_2(a,r)$ and $K_D(a,r)$ defines the balls around $a \in M$ with radius $r$, wrt. the three metrics. Show, that for $0<r$ and $0<s$ it is true that
$$K_D(a,\min\{r,s\})\subseteq K_1(a,r)\cap K_2(a,s)\subseteq K_D(a,\max\{r,s\}).$$
ii) Let $A\subseteq M$. Show that, if $A$ is open wrt. to at least one of the metrics $d_1$ and $d_2$, then it is also open wrt. to $D$.
$\textbf{Remark}$: Can I use the duality of open and closed sets?
iii) Assume. $A\subseteq M$ is compact wrt. $D$. If so, show that it is also compact wrt. both $d_1$ and $d_2$.
We assume that a ball around a point $a\in M$ with radius $r$ consists of all points $x\in X$ such that the distance from $a$ to $x$ is less than $r$ (the case when it is less or equal is considered similarly). To prove (i) note that if $x\in K_D(a,\min\{r,s\})$ then $d_1(a,x)<r$ and $d_2(a,x)<s$, so $x\in K_1(a,r)\cap K_2(a,s)$. Also if $x\in K_1(a,r)\cap K_2(a,s)$ then $d_1(a,x)<r\le\max\{r,s\} $ and $d_2(a,x)<s\le\max\{r,s\}$, so $D(a,x)=\max\{ d_1(a,x), d_2(a,x)\}< \max\{r,s\}$ and so $x\in K_D(a,\max\{r,s\})$. The above implies that the identity maps from $(M,D)$ to $(M,d_1)$ and $(M, d_2)$ are continuous. Then (ii) follows because $A$ in $(M,D)$ is a preimage of a open set $A$ in $(M, d_i)$ with respect to a continuous map. Also (iii) follows because $A$ in $(M,d_i)$ is an image of a open set $A$ in $(M, D)$ with respect to a continuous map.