My question is more regarding the induction process and I want to make sure I'm not making a build-up error in the proof.
Proof by induction on $k$, the number of edges. Let $G$ be a graph with $n$ vertices.
Base case: for $k = 0$ the claim holds as we have $n$ isolated vertices and thus, $n - 0$ connected components.
Inductive Hypothesis: suppose the claim holds for all $j\geq0$. That is every graph with $n$ vertices and $j$ edges has at least $n - j$ connected components.
*Inductive Step: Suppose we now have $j + 1$ edges. We know that for $j$ edges the claim holds from the inductive hypothesis. If we add an edge to the graph from the inductive hypothesis then we have the following cases:
1)We connect two components thus decreasing the number of connected components by $1$. Then the number of connected components is $n - j - 1 = n - (j +1)$.
2)We add an edge within a connected component, hence creating a cycle and leaving the number of connected components as $ n - j \geq n - j - 1 = n - (j+1)$.*
In either case the claim holds, therefore by the principle of induction the claim is true for all graphs.