A simple graph $G=(V,E)$ is a forest iff the number of components is $|V|-|E|$.
Let $|V| = n$ and $|E| = k$.
$\Rightarrow$ How can I use the fact that a simple graph on $n$ vertices and $k$ edges has at least $n-k$ components (I actually have the proof of this)? Do we have equality since adding more edges forms a cycle?
$\Leftarrow$ For the converse, if the graph has $|V|-|E|$ components, then each of the components should be a tree because each $n_i$-vertex tree contains $n_i-1$ edges. Is this a good enough argument?