how can I proof that?

from the writtings of rabbi mosa ibn maimon stating that prophet Mohammed was an Ishmailite:

All those words of Jesus of Nazareth and of this
Ishmaelite
who arose after him are only to make straight the path for the messianic king and to prepare the whole world to serve the Lord together. As it is said: 'For then I will change the speech of the peoples to a pure speech so that all of them shall call on the name of the Lord and serve him with one accord' (Zephaniah 3:9

A. James Rudin
.
Christians & Jews Faith to Faith: Tragic History, Promising Present, Fragile Future, Jewish Lights Publishing, 2010, pp. 128–129

2.
in the jewish book the secrets of Rabbi Shimon :

ince he saw the kingdom of Ishmael that was coming, he began to say: "Was it not enough what the wicked kingdom of Edom has done to us,

but [we deserve] the kingdom of Ishmael too?

" At once Metatron, the foremost angel (sar ha-penim), answered him and said: "Do not fear, son of man,

for the Almighty only brings the kingdom of Ishmael

in order to deliver you from this wicked one (Edom).

He raises up over them (Ishmaelites) a prophet according to His will and He will conquer the land for them

, and they will come and restore it to greatness, and a great dread will come between them and the sons of Esau." Rabbi Simon answered him and said: "How [is it known] that they are our salvation?" He (Metatron) said to him:

"Did not the prophet Isaiah say that 'he saw a chariot with a pair of horsemen etc.'? Why did he put hte chariot of asses before the chariot of camels when he should rather have said 'a chariot of camels and [then] a chariot of asses,' because when he (Ishmael, i.e. the Arabs) goes forth [to war], he rides upon on a camel, and when the kingdom will arise by his hands he rides upon an ass? [

Given that he said the reverse of this], the chariot of asses, since he (the Messiah) rides upon an ass,

shows that they (the Ishmaelites, represented by the chariot of camels

) are a salvation for Israel, like the salvation of the rider on an ass (i.e. the Messiah).

3. texts mentioning that the followers of the prophet are arabs sons of ishmael fro the first 100 years of hijra:

First : Sebeos the bishop of the Bagratunis writting in 40 AH:

At that time a certain man from along those same sons of Ismael, whose name was Mahmet[i.e., Muḥammad], a merchant, as if by God's command appeared to them as a preacher [and] the path of truth. He taught them to recognize the God of Abraham, especially because he was learnt and informed in the history of Moses. Now because the command was from on high, at a single order they all came together in unity of religion. Abandoning their vain cults, they turned to the living God who had appeared to their father Abraham. So, Mahmet legislated for them: not to eat carrion, not to drink wine, not to speak falsely, and not to engage in fornication. He said: 'With an oath God promised this land to Abraham and his seed after him for ever. And he brought about as he promised during that time while he loved Ismael. But now you are the sons of Abraham and God is accomplishing his promise to Abraham and his seed for you. Love sincerely only the God of Abraham, and go and seize the land which God gave to your father Abraham. No one will be able to resist you in battle, because God is with you

Second: The chronicle of Khuzistan written 40 AH by Nestorian christian:

Then God raised up against them the sons of Ishmael, [numerous] as the sand on the sea shore, whose leader (mdabbrānā) was Muḥammad (mḥmd). Neither walls nor gates, armour or shield, withstood them, and they gained control over the entire land of the Persians. Yazdgird sent against them countless troops, but the Arabs routed them all and even killed Rustam. Yazdgird shut himself up in the walls of Mahoze and finally escaped by flight. He reached the country of the Huzaye and Mrwnaye, where he ended his life. The Arabs gained countrol of Mahoze and all the territory. They also came to Byzantine territory, plundering and ravaging the entire region of Syria. Heraclius, the Byzantine king, sent armies against them, but the Arabs killed more than 100,000 of them

Third: in the teaching of jacob a 7th century anti jewish document :

When the candidatus was killed by the Saracens, I was at Caesarea and I set off by boat to Sykamina. People were saying "the candidatus has been killed," and we Jews were overjoyed. And they were saying that the prophet had appeared, coming with the Saracens, ........., and I said to him: "What can you tell me about the prophet who has appeared with the Saracens?"

Source : Doctrina Jacobi V.16, 209. [p. 57]

This was written between 634 and 640 in the conquest of palestine , Saracens means arabs , it is used by some christians and Jews to refer to the arabs as a reference to their mother Hagar being exiled by Sara

4. From the bible :

"And as for Ishmael, I have heard you. Behold, I have blessed him, and will make him fruitful, and will multiply him exceedingly. He shall beget twelve princes, and I will make him a great nation" (Genesis 17:20)

Now this is the genealogy of Ishmael, Abraham's son, whom Hagar the Egyptian, Sarah's maidservant, bore to Abraham. And these were the names of the sons of Ishmael, by their names, according to their generations: The firstborn of Ishmael, Nebajoth; then Kedar, Adbeel, Mibsam, Mishma, Dumah, Massa, Hadar, Tema, Jetur, Naphish, and Kedemah. These were the sons of Ishmael and these were their names, by their towns and their settlements, twelve princes according to their nations 17These are the years of the life of Ishmael, one hundred and thirty-seven years; and he breathed his last and died, and was gathered to his people.18They settled from Havilah to Shur which is east of Egypt as one goes toward Assyria; he settled in defiance of all his relatives (Genesis 25:12-18).

Passage 18 shows the geographic land of the Arabs

The oracle about Arabia. In the thickets of Arabia you must spend the night, O caravans of Dedanites. 14Bring water for the thirsty, O inhabitants of the land of Tema, Meet the fugitive with bread.15For they have fled from the swords, From the drawn sword, and from the bent bow And from the press of battle.16For thus the Lord said to me, "In a year, as a hired man would count it, all the splendor of Kedar will terminate; 17and the remainder of the number of bowmen, the mighty men of the sons of Kedar, will be few; for the LORD God of Israel has spoken."
Isaiah 21

This prophecy is talking about Arabia , where it describes the sons of Kedar (who is the son of Ishmael and the grandfather of the prophet) and they live in Arabia , they are arabs

This prophecy also describes the hijra to Madina and the battle of Badr

5. From the hadeeth :

Wathila b. al-Asqa' reported:

I heard Allah's Messenger (ﷺ) as saying: Verily Allah granted eminence to Kinana from amongst the descendants of Isma'il and he granted eminence to the Quraish amongst Kinana and he granted eminence to the Quraish amongst Banu Hashim and he granted me eminencece from the tribe of Banu Hashim.

Saheeh Muslim Book 30, Hadith 5653

ِAnother example :

Narrated Salama bin Al-Akwa`:

The Prophet (ﷺ) passed by some people of the tribe of Bani Aslam who were practicing archery. The Prophet said, "O Bani Isma`il ! Practice archery as your father Isma`il was a great archer. Keep on throwing arrows and I am with Bani so-and-so." So one of the parties ceased throwing. Allah's Apostle said, "Why do you not throw?" They replied, "How should we throw while you are with them (i.e. on their side)?" On that the Prophet (ﷺ) said, "Throw, and I am with all of you."

Saheeh Bukhari
Vol. 4, Book 52, Hadith 148