Then Allah says:

سَنُرِيهِمْ آيَاتِنَا فِي الْآفَاقِ وَفِي أَنفُسِهِمْ ...

We will show them Our signs in the universe, and within themselves,

meaning, `We will show them Our evidence and proof that the Qur'an is true and has indeed been sent down from Allah to the Messenger of Allah, through external signs, فِي الْآفَاق (in the universe),' such as conquests and the advent of Islam over various regions and over all other religions.

Mujahid, Al-Hasan and As-Suddi said,

"And signs within themselves means, the battle of Badr, the conquest of Makkah, and other events which befell them, where Allah granted victory to Muhammad and his Companions, and inflicted a humiliating defeat on falsehood and its followers.''

It may also be that what is meant here is man and his physical composition. As detailed in the science of anatomy, which indicates the wisdom of the Creator -- may He be blessed and exalted -- and the different inclinations and opposite natures that people have, good and bad, etc., and the (divine) decrees to which man is subjected, having no power to change and over which he has no control.

Allah's statement,

... حَتَّى يَتَبَيَّنَ لَهُمْ أَنَّهُ الْحَقُّ أَوَلَمْ يَكْفِ بِرَبِّكَ أَنَّهُ عَلَى كُلِّ شَيْءٍ شَهِيدٌ ﴿٥٣﴾

until it becomes manifest to them that this (the Qur'an) is the truth. Is it not sufficient in regard to your Lord that He is a Witness over all things?

means, sufficient is Allah as a Witness over the words and deeds of His servants, and He bears witness that Muhammad is telling the truth in what he says, as Allah says:

لَّـكِنِ اللَّهُ يَشْهَدُ بِمَآ أَنزَلَ إِلَيْكَ أَنزَلَهُ بِعِلْمِهِ

But Allah bears witness to that which He has sent down unto you; He has sent it down with His knowledge. (4:166)