Prove: If a connected graph $K$ has $n$ vertices and $x$ edges with $x > n − 1$, will it contain a cycle?

Question

I'm new to graph theory and I'm trying to prove this problem, but I am not sure how to go about it.

Answer 1

Let us assume it doesn't, so it's acyclic. So it is either a forest or a tree, which has $n-1$ edges. However it has at least $n$ edges, so it can't be neither. If it can't by an acyclic graph, it has to have a cycle, therefore it has a cycle

Answer 2

Note that with each edge you add to the graph that does not generate a cycle, the number of connected components decreases by exactly one.