Do Turing machines assume something infinite at some point?
A Turing machine does not have the ability "to compare numbers of arbitrary size through the &lt;= comparator" because a Turing machine does not have a "&lt;= comparator". A Turing machine has a fixed, finite set&nbsp;$Q$ of states and a fixed, finite tape alphabet&nbsp;$\Sigma$. At each step of the computation, the Turing machine looks at its current state and the symbol under its read/write head and decides what to do next: which state to enter, which symbol to write to the tape and which way to move the tape head.

Because of this, a Turing machine cannot compare arbitrarily large numbers in a single &lt;= instruction. Using the state, it can remember at most $|Q|$ different numbers and, using the alphabet, it can write at most $|\Sigma|$ different numbers in a single tape cell (using each possible symbol to represent one number).  As such, to compare arbitrarily large numbers on a Turing machine, you must write each number as a sequence of digits on the tape and write an algorithm that will take multiple steps to compare those two numbers.  As you can imagine, this makes writing Turing machine programs a rather fiddly endeavour.

Turing machines don't really "deal with infinity": they deal with unbounded finite things, at least in their standard definition.  The input is a finite string and, after any finite number of steps, the machine has only examined or written to a finite number of tape cells. There is no bound on the size of the input or the number of computation steps but, the input is finite and, after any finite number of steps, only a finite amount of output has been produced.