Every graph with $\delta(G) \ge 2$ has a cycle of length at least $\delta(G)+1$?
If $\delta(G) \geq 2$ every vertex is connected to at least 2 others. This invariably leads to a cycle in any finite graph.

To see why, try to construct a path without a cycle from a graph with $\delta(G) \geq 2$. Every vertex you add is connected to either a previously added vertex (forming a cycle), or an other vertex. However, in turn, that vertex is connected... Since the graph is finite you will at some point run out of vertices to add which you haven't seen already, forcing you to form a cycle.