Learn English – Multiplication: names of some mathematical symbols

Yes, "to" is the correct word here. You are "generalizing the theorem to the larger set".

In English we talk about "direct objects" of a verb and "indirect objects". In general the direct object is the thing actually acted on, and the indirect object is a destination of some sort. The exact meaning of a direct versus an indirect object depends on the verb. We normally put the direct object immediately after the verb, and any indirect objection preceded by the word "to".

The common type of example I always see is, "I gave the book to Bob". The direct object is "book"; the indirect object is "Bob". The book is the thing given; Bob is the destination of that giving.

Slightly different: "I explained the problem to Carl." The thing I explained is the problem. The person receiving the explanation is Carl.

In your case, "This theorem generalizes Theorem 1 to real numbers", the direct object is Theorem 1. It is the thing being generalizes. The indirect object is real numbers. They are the target or destination of the generalization.

• The absolute value of the difference between S and
• the sum from i equals one through i equals n of
• the function f evaluated at t sub i times the width of each i
• is less than epsilon.

If it is clear that i and n are one-indexed, then "the sum from i equals one through i equals n" can be replaced by "the sum of the first n terms". "The width of each i" is an interpretation of "delta i".

• The function is one divided by the quantity x plus one close quantity, all divided by the square root of x.
• The integral from zero to infinity of the function d x
• equals the limit as s goes to zero of the integral from s to 1 of the function d x
• plus the limit as t goes to infinity of the integral from 1 to t of the function d x.

"Goes to" can be replaced by "goes toward", or (as Damkeng suggests) "tends to", or (as J.R. suggests) "approaches".

I often use a notation like "integral from x equals a to x equals b" instead of "integral from a to b". I also often say "to positive infinity" instead of "to infinity".

I "factored out" the definition of the function in the first sentence. If the function were easy to say (such as "x squared"), I would not "factor out" the definition of the function. Instead, I would include the "x squared" in the statements of the integrals, a la Damkeng's answer.