Zooming in on a photo
Zoom into a digital photo and each doubling reveals a little less new detail until the image settles and stops changing. A convergent sequence behaves the same way: its terms crowd in around one limit value and stay there.
Getting arbitrarily close
A sequence (aₙ) converges to L if its terms get and stay arbitrarily close to L. Formally: for every ε > 0 there is an N such that |aₙ − L| < ε for all n > N.
Bounded vs convergent
Every convergent sequence is bounded, but not every bounded sequence converges (e.g. (−1)ⁿ oscillates).