How to prove for every positive integers $k$ if the degree set of vertices of graph is $( 1 , 1 , 2 , 2 , 3 ,3 , .... , k , k )$ , we can create a graph .
For example if $k = 3$ we can have a graph that has $6$ vertices that $\text{deg}(v_1)$ and $\text{deg}(v2)$ is $1$ , $\text{deg}(v_3)$ and $\text{deg}(v_4)$ is 2 and $\text{deg}(v_5)$ and $\text{deg}(v_6)$ is 3.
This is what I tried :
I could understand that in every case we have $2k$ vertices. And I think it can be solved by induction. We know that $p(1)$ is true that means it is possible to create a graph with $2$ vertexes with degree of $1$ for each of them. actually I do not know how to prove for $p(k+1)$ please help me solve this proof in this way or another way.
Worth a thousand words, they say.