find a sequence of closed connected subsets $V_n$ of $\mathbb R^2$ s.t. $V_n\supseteq V_{n+1}$ and $\cap^{\infty}_{i=1}V_i$ is not connected

Find an example of a sequence $V_n$ of closed and connected subsets of the Euclidean plane satisfying $V_n \supseteq V_{n+1}$ such that $\cap_{i=1}^{\infty}V_i$ is not connected.

Normally I would show how far I got on my own but I can't figure this one out, I have been thinking of different pictures in my head to come up with an idea but I can't. If anyone could give me a tip or an example that would be great. Thanks in advance.

Consider two disjoint half-planes connected by a "bridge" that goes each time farther towards infinity, something like $$V_n = \{ (x,y) \in \mathbb{R}^2 : |x| \ge 1 \} \cup \mathbb{R} \times [n, + \infty).$$
Then $V_n$ is closed (union of two closed sets) and connected (it's even path connected), but $\bigcap_n V_n$ is the disjoint union of two half planes and is disconnected.
• @nik Thanks for your answer. So we would have something like $V_n = \{ (x,y)\in \mathbb R^n: |x| \geq 1 \}\cup \{ (x,y) \in \mathbb R^2 : |x| \leq 1, y\geq n \}$? The first set being the two closed sides of the plane and the last set being the bridge? – Slugger Apr 2 '14 at 14:30