Draw a circle. Now, outside of that circle put the pen to your imaginary (or real) paper and start randomly moving around until you enter that circle, after which you're free to keep moving around randomly but only within that circle.
Once you've done that, draw a smaller circle somewhere inside the circle you're now trapped in, keep moving until you hit that circle, and now keep moving but be confined to that one.
Repeat this forever, drawing ever smaller circles, indefinitely closing in on a point. The moving path not only keeps getting closer to that point, but it does so with absolute reliability: once it gets within a certain range of it, it never goes back out of that range.
Now, imagine that every distance, no matter how small, your path moves, it gets caught in one of the concentric circles. Now it is always approaching the final point no matter what. This is what it means to converge.