Problem #PRU-5087

Problem

Let’s prove the following statement: every graph without isolated vertices is connected.
Proof We use the induction on the number of vertices. Clearly the statement is true for graphs with $2$ vertices. Now, assume we have proven the statement for graphs with up to $n$ vertices.
Take a graph with $n$ vertices by induction hypothesis it must be connected. Let’s add a non-isolated vertex to it. As this vertex is not isolated, it is connected to one of the other $n$ vertices. But then the whole graph of $n+1$ vertices is connected!