Why does ionisation energy increase across a period?

As you go across the period, the nuclear charge (or number of protons increase) but the shielding (or the number of shells) stays the same as all of the elements are on the same period. This causes the nuclear attraction between the nucleus (that is positive) and outermost electron (that is negative) to increase, so the distance between them, also called the atomic radius, decreases.This attraction makes it harder to lose that electron and so the energy to remove it (the ionisation energy) increases as you go across the period