if a graph G(V,E) is connected $|E|\geq|V|-1$
Imagine a process in which you start with the empty graph, and add the edges one by one. Keep track of the number of connected components. In the beginning, there are $|V|$ connected components. Each edge you add either belongs inside a connected component, or merges two connected components, and in any case, the number of connected components decreases by at most 1. Hence after adding $|E|$ edges, the number of connected components is at least $|V| - |E|$. If the graph is connected, $|V| - |E| \leq 1$, and so $|E| \geq |V| - 1$.