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\begin{align*} 2|E| = \sum_{v \in V} \mathrm{deg}(v)) \geq 2|V|-1 &\Rightarrow |E| \geq |V| - \frac{1}{2} \\ &\Rightarrow |E| \geq |V| \\ &\Rightarrow (v,E) \mathrm{is \; not \; a \; tree} \end{align*}