This paper presents an approach for visible light communication-based indoor positioning using compressed sensing. We consider a large number of light emitting diodes (LEDs) simultaneously transmitting their positional information and a user device equipped with a photo-diode. By casting the LED signal separation problem into an equivalent compressed sensing framework, the user device is able to detect the set of nearby LEDs using sparse signal recovery algorithms. From this set, and using proximity method, position estimation is proposed based on the concept that if signal separation is possible, then overlapping light beam regions lead to decrease in positioning error due to increase in the number of reference points. The proposed method is evaluated in a LED-illuminated large-scale indoor open-plan office space scenario. The positioning accuracy is compared against the positioning error lower bound of the proximity method, for various system parameters.