Let $G$ be a connected graph.
a) Show that if $T$ is a spanning tree of $G$ that is distance preserving from some vertex of $G$, then $diam(G) \leq diam(T) \leq 2diam(G)$
b) show that for every positive integer $a$, there exist a connected graph $G$ of diameter $a$ and vertex $v$ of $G$ such that for every integer $b$ with $a\leq b\leq 2a$, there is a spanning tree $T$ that is distance preserving from $v$ and $diam(T)=b$.
Here is what I got so far.
Let $T$ is a spanning tree of $G$ that is distance preserving from some vertex of $G$. Let $u \in V(T)$. Since $T$ is spanning tree of $G$, $V(T) = V(G)$. And since $T$ is distance preserving from some vertex of $G$, there is some vertex $v\in V(G)$ such that $d_T(u,v)=d_G(u,v)$. This is where I'm stuck, these info don't tell me anything about the diameter, or maybe they do, but I just missed it.
For (a): let $v1,v2$ be vertices such that $d_G(v1,v2)=diam(G)$. Then $diam(T) \ge d_T(v1,v2) \ge d_G(v1,v2) = diam(G)$; the first inequality is obvious from the definition of diameter, while the second inequality holds for any subgraph $T$ of $G$, not just spanning trees.
As for the other inequality, let $v$ be the chosen vertex from which $T$ is distance-preserving. Letting $v1,v2$ be vertices such that $d_T(v1,v2)=diam(T)$, we have $$ diam(T) = d_T(v1,v2) \le d_T(v1,v) + d_T(v,v2) = d_G(v1,v) + d_G(v,v2) \le diam(G) + diam(G); $$ the first inequality is the triangle inequality.