Degree sequence multigraph without loop I am trying to prove the following

$d_n$ is a degree-sequence of the multigraph  $G$ (without loops) if and only if $\sum_{i=1}^{n}{d_i}$ is even and $d_1 \le\sum_{i=2}^{n}{d_i}$

I have no problems showing the first part.
By the "handshake"-lemma $\sum_{i=1}^{n}{d_i}=2e$ where $e$ is the number of edges in $G$
$$d_1 \le e $$
$$d_1\le \frac{\sum_{i=1}^{n}{d_i}}{2}$$
Hence $d_1 \le \sum_{i=2}^{n}{d_i}$
However, I don't know how to show the other part. Would appreciate some tips or hints
 A: Here's the construction I've found. As far as I know, the result was first proven by Hakimi in 1962 (check the article On Realizability of a Set of Integers as Degrees of the Vertices of a Linear Graph).
Theorem. Let $d_1\leq...\leq d_n$ be a sequence of natural numbers. This sequence can be realized by some multigraph iff $S=d_1+...+d_n$ is even and $d_n\leq S/2$.
Proof. The necessary part follows from the handshaking lemma. Let's prove that the conditions are also sufficient.
If $n=1$, the only possibility is $d_1=0$. This case is trivially realizable, thus suppose $n\geq2$. Consider $n$ vertices $v_1,...,v_n$. Connect $v_1$ and $v_2$ with $d_1$ edges, $v_2$ and $v_3$ with $d_2-d_1$ edges, $v_3$ and $v_4$ with $d_1-(d_2-d_1)$ edges, etc. You'll get a graph $G$ of the following form (with the degrees subscribed):

$G$ is almost corresponding to our sequence: we have to add $d_n-x$ new edges to $v_n$ to complete the construction. This addition is possible if $G'=G-v_n$ contains enough edges, then you may cut some edge from $G'$ in two parts connecting these parts with $v_n$. The procedure will obviously preserve the degrees of $v_1,...,v_{n-1}$ and will increase $d(v_n)$ by $2$:

Now, we have to prove that $G'$ contains at least $(d_n-x)/2$ edges (it's easy to see that $d_n-x$ is even). If this is not the case, then $|E(G')|=(d_1+...+d_{n-1}-x)/2<(d_n-x)/2$. This implies $d_1+...+d_{n-1}<d_n$ and $S/2<d_n$ giving us a contradiction $\blacksquare$
Let's show how the described method works applied to $2,4,6,8$:

