In my analysis class I use the following problem to illustrate the divergence of the harmonic series (consider this as a hint for solving it).
Exercise. A beetle creeps along a 1-meter infinitely elastic tape with constant velocity. Every hour the tape is lengthened out by 1 meter, and the beetle remains at the same rate of the tape it has already reached to the moment. Will the beetle ever reach the end of the tape?
This is not a paradox but a calculation of a mathematically idealised model "from life". Do you have some other, probably nicer examples which illustrate some standard but deep results in analysis, algebra, probability, geometry, and so on, and so on, and so on. Please keep in mind an average undergraduate student as the audience for your example and allow others to use it in his/her teaching.