Let V be a finitely-generated vector space over k and let S be a (possibly infinite) spanning set for V. Show that there exists a finite subset of S that also spans V.

You are standing next to a really big circular lake. you want to measure the diameter of the lake, but you don't want to have to swim across with a measuring tape! you decide to walk around the perimeter of the lake and measure its circumference, and find that it's 400\pi\text{ m}400? m. what is the diameter dd of the lake?