How do I prove this by contradiction?
A bridge in a connected graph is an edge whose removal disconnects the graph. Let $n \ge 1$.
Prove that a connected $2n$-regular graph has no bridges.
So far what I have is this:
Suppose for contradiction that a $2n$-regular graph has a bridge $uv$. By removing the edge $uv$, there is now $2$ connected graphs $A$ and $B$. Since $u,v$ has more than $2n$ vertices in the original graph, both $u$ and $v$ now have $2n -1$ vertex degrees.
The picture below is what I believe is a $2n$-regular graph