Proving that A connected graph on $n$ vertices is a tree iff it has $n-1$ edges
Solution 1:
Hint: Argue that a connected graph with $n$ vertices and $n - 1$ edges cannot have a cycle and is thus a tree.
And yes, your induction looks fine, although you seem to be restricting yourself to a leaf. Maybe you can try a more general case by detaching the new vertex and considering the two disconnected components (both of which will satisfy the inductive hypothesis).