Wait what? (1+2+3+4+...

[BAndrew]

@Naked? No
A good example is the integral. For instance with an integral you can calculate the area of a function exactly by calculating the sum of infite infinitesimals I am not going into details into how that works. You can read here, but it requires some more advanced mathematical knowledge.

@Bridge

By rejecting the use of infinitesimals and infinity in calculations you are rejecting a whole branch of mathematics which by the way works just perfectly.

But if you feel so strongly about it, prove to me that "the limit of 1/n as n approaches infinity is 0". Without using limits or algebra or anything which is designed for real numbers.

If I have to prove something I have to start from somewhere (axioms) don't you think? The definition and explanation of a limit is the following (from wikipedia):

Suppose f is a real-valued function and c is a real number. The expression



means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, the above equation can be read as "the limit of f of x, as x approaches c, is L".

Augustin-Louis Cauchy in 1821,[2] followed by Karl Weierstrass, formalized the definition of the limit of a function as the above definition, which became known as the (ε, δ)-definition of limit in the 19th century. The definition uses ε (the lowercase Greek letter epsilon) to represent a small positive number, so that "f(x) becomes arbitrarily close to L" means that f(x) eventually lies in the interval (L - ε, L + ε), which can also be written using the absolute value sign as |f(x) - L| < ε. The phrase "as x approaches c" then indicates that we refer to values of x whose distance from c is less than some positive number δ (the lower case Greek letter delta)—that is, values of x within either (c - δ, c) or (c, c + δ), which can be expressed with 0 < |x - c| < δ. The first inequality means that the distance between x and c is greater than 0 and that x ≠ c, while the second indicates that x is within distance δ of c.

Note that the above definition of a limit is true even if f© ≠ L. Indeed, the function f need not even be defined at c.

If n-->infinity then we define:

The sequence a(n) has a limit L ∈ ℝ and we write

lim a(n) = L if for every ε>0, exists a n0 ∈ ℕ* such that n>n0 then
n-->∞

|a_n - L| < ε

It is really Math heavy going. If you don't accept the definition of course then I can't prove anything