In many situations, for example in teaching convergence to students not specialising in mathematics, an approach to the idea of a limit is required which is easily motivated, mathematically acceptable, and yet not technically terrifying. The usual outcome is a compromise which lies uneasily between these aims. Here, I try to present an introduction to convergent sequences (of real numbers) which meets these objectives; it can easily be adapted to accommodate limits of functions. The approach is based on the sandwich rule (or squeeze principle) and the intuitively straightforward idea of the limit of a monotone sequence.