In one of my lectures to a group of Discrete Mathematics students (all of which are CS majors, not math), I go over the proof that the cardinality of the continuum is strictly larger in size than the cardinality of the naturals; ie the reals are uncountable.
This lesson closes out our disscussion on generalized functions, and I found students understand the proof of Cantor's argument fairly well. However, this proof only shows that there exists at least one real outside any bijective function, when in fact nearly all reals are excluded.
As stated here
Which reals could possibly be “missing” from our universe? Every real you can name—42, π, √e, even uncomputable reals like Chaitin’s Ω—has to be there, right? Yes, and there’s the rub: every real you can name. Each name is a finite string of symbols, so whatever your naming system, you can only ever name countably many reals, leaving 100% of the reals nameless.
Or did you think of only the rationals or algebraic numbers as forming a countable dust of discrete points, with numbers like π and e filling in the solid “continuum” between them? If so, then I hope you’re sitting down for this: every real number you’ve ever heard of belongs to the countable dust! The entire concept of “the continuum” is only needed for reals that don’t have names and never will.
Which is incredible, and I really want to convey this to my students as well, however I don't know of a good proof or argument that is at their level (so no measure theory) to show that "100%" of the real numbers are excluded from any possible function from $\mathbb{N}$
Clearing up my question so that it's easy to see exactly what I am asking for:
Can it be proved using only basic set theory, functions, and calculus that "100%" of all real numbers are excluded from any function $f:\mathbb{N}\rightarrow\mathbb{R}$?
Although measure theory is of course needed to do this formally, perhaps there's a way to discuss this with your students informally.
You could argue like this.
List the numbers: $x_1 = f(1)$, $x_2 = f(2)$, $x_3 = r(3)$,...
Draw a sketch of the first few on a number line; it doesn't matter exactly where you put them.
Surround $x_1$ with an interval labelled $I_1$ of total length $1/2$, going from $x_1-1/4$ to $x_1 + 1/4$.
Next, surround $x_2$ with an interval labelled $I_2$ of togal length $1/4$, going from $x_2 - 1/8$ to $x_2 + 1/8$. So, the total length of $I_1$ and $I_2$ is at most $\frac{1}{2} + \frac{1}{4} = \frac{3}{4}$.
And now continue. At the $n$th stage, you'll have intervals $I_1,...,I_n$ whose lengths total up to $1 - \frac{1}{2^n}$.
And so, even if you continue letting $n$ go to infinity and take all the intervals all together, their lengths will total up to at most $1$! How can that possibly cover the entire real number line, which has infinite length?
If you wanted to, you could repeat this, except that you can start with $I_1$ having length $.00005$ instead of $.5$; the total lengths will total up to at most $.0001$!