discrete mathematics - Inductively the number of edges for a connected graph

Claim: Every connected graph with $n$ vertices has at least $n - 1$ edges.

Proof by induction:

$P(n)$ = "Every connected graph with $n$ vertices has at least $n - 1$ edges." $\forall n \ge 1$

Base case: A graph with $1$ vertex has $0$ edges.

Inductive step:

1. Remove a vertex with the lowest degree and all its edges from the connected graph with $n + 1$ vertices.




2. The resulting graph has $n$ vertices. Apply the inductive hypothesis - this graph has at least $n - 1$ edges.




3. When you add the removed vertex back, it must connect to at least $1$ vertex in order for the overall graph to be connected.




4. The graph now contains at least $n - 1 + 1$ or $n$ edges

I'm new to graph theory, so I'm wondering if I completed this induction proof with the right steps. Any other methods or additions to this proof would also be appreciated. Thanks!