Are you familiar with accusativus cum infinitivo (ACI)?
All of the bolded infinitives in your quote belong to an ACI structure, and should not be treated in isolation from other parts of the structure.
As the name suggests, ACI comes with an accusative and an infinitive.
In English it is often best translated as a subordinate clause starting with "that", and it can be regarded as a form of indirect speech.
I think it is best not to think of ACI as indirect speech, but as a peculiar syntactical construction in Latin.
The use of ACI has nothing to do with the fact that you are within a direct quote.
The structure is typically governed with a verb of seeing, hearing, thinking, believing, or some other such form of observation (in a broad sense).
Also other kinds of verbs are possible, like speaking.
Judging by the frequency, I believe this segment in your textbook introduces ACI, and I hope the structure is explained in the material.
For a very simple first example, consider this sentence: Audio te canere.
The ACI is te canere; the verb is in infinitive and the subject te is in accusative.
This ACI is the object of the verb audio.
The whole sentence can be translated as "I hear that you sing" or "I hear you sing".
The tense of the infinitive indicates when the event happened with respect to the observation.
Video te cecinisse is "I see that you have sung".
Every ACI has the three main parts: the actual accusative and infinitive, and the governing verb.
I will point these parts out in your passage, stripped of all other structure.
Sometimes the governing verb or accusative is not explicitly there due to ellipsis to avoid repetition.
- nolite arbitrari me fore — "do not think that I will be"
- videbatis eum esse in hoc corpore — "you saw that it was in this body"
- eundem esse credite — "believe that it is"
- persuaderi potuit animos vivere — "could persuade me that souls live"
- persuaderi potuit animos mori — "could persuade me that souls die"
To quote someone verbatim, you can do as is done in your quote.
Start with "He said:" and tell what he said.
The Romans did not have quotation marks, but they did certainly have quotes.
The ACI can also be used to quote someone (although not verbatim).
Consider these two ways to quote:
Ille dixit: Marcus abest. — He said: Marcus is away.
Ille dixit Marcum abesse. — He said that Marcus is/was away.