Why does the first ionisation energy increase across period 3?

The 1st ionisation energy is defined as the energy required to remove the outermost electron from one mole of gaseous atoms, forming one mole of gaseous 1+ ions. Moving from left to right the proton number increases across a period, therefore the nucleus becomes more positively charged. This increasing positive charge creates greater nuclear attraction between the positively charged nucleus and the negatively charges outermost electrons. Therefore more energy is required to remove these electrons from the atom. In the third period all of the outermost electrons occupy the third shell so electron shielding is constant and will not affect the ionisation energy. Referring back to the Definition of 1st ionisation, if the energy to remove the outermost electron increases then so to does the 1st ionisation energy.