Showing that graph of spanning trees are connected
Use induction on the number of edges and vertices in $G$.  Fix some edge $e$ of $G$.  If $S$ and $T$ are two different spanning trees that do not use $e$, apply the induction hypothesis on the graph $G\setminus e$.  If both use $e$, apply the inductive hypothesis to the contracted graph $G/e$.  If only one does, w.l.o.g., $S$, replace $T$ by $T'$, obtained by adding $e$ to $T$ and removing any edge in the (unique) cycle in $T+e$.  Either $T'$ is $S$, or we are back in a previous case.