As part of a homework assignment, I am doing a proof for a generalised variant of Karger's algorithm and am stuck at a particular step. I have proven that for a graph $G=(V,E)$ [writing $n=|V|$] with a minimum $k$-cut $C$, $$|C| \leq \sum\limits_{l=1}^{k-1} d_l$$ where $d_1\leq d_2\leq\ldots$ are the degress of $v_1,v_2,\ldots\in V$. I need some way of estimating an upper bound on $\frac{|C|}{|E|}$ in terms of $n$ and $k$ alone. For instance in the case of minimum two-cuts (what are normally just called "minimum cuts") we can show $$|C| \leq d_1 \leq \frac{1}{n}\sum\limits_{l=1}^n d_l = \frac{2|E|}{n} \Rightarrow \frac{|C|}{|E|} \leq \frac{2}{n}$$
Any help would be appreciated (there is a strong possibility of an "XY problem"-situation here but please give it a go anyways).