In my opinion Baby Rudin isn't a great textbook to learn from. It's the standard textbook for analysis, but I think it's better as a reference than as something to learn from. (In contrast, I found Hatcher to be a great textbook for learning algebraic topology.) I have heard that Terry Tao's analysis textbook is good, but I haven't read it myself so I can't confirm.

So let there be the metric space (X, d), let C(N, R) be the set of all convergent sequences in R and let there be the function f: C(N, R) |--> X; f(x_n) --> lim x_n. There's two situations: one where the distance associated with C(N, R) is d_infinity, which is the maximum difference for all elements in the two sequences in C(N, R), and one where the distance associated with C(N, R) is d(x, y) = 1/n where n is the smallest k where x_k != y_k. I need to prove that f is continuous in the first situation and not continuous in the second situation.

Imma make a glossary in case someone wants to help but doesn't know the terminology.

• Metric space: it's a set associated with a function d that fulfills the properties of non-negativity, commutativity and triangular inequality. The function d is called the distance.

• Distance: it's the function d associated with a metric space.

• Sequence: an enumerated collection of objects. Like a set, but order matters. The elements of a sequence x_n are denominated by their order x_1, x_2, x_3, and so on

• Convergence: a sequence is said to be convergent when it has a limit.

• Limit of a sequence: a is the limit of a sequence when for every epsilon > 0 there's a natural number n0 such that for every n > n0 you have |x_n - a| < epsilon. If a sequence doesn't have a limit, it's called a divergent sequence.

• Continuity of a function in metric spaces: a function f: X |--> Y is said to be continuous when for every epsilon > 0 there's a delta > 0 such that d1(x, y) < delta implies d2(f(x), f(y)) < epsilon. In this example, x and y are any random elements of X and d1 and d2 are the distances associated with the metric spaces X and Y.

That's pretty cool and impressive man. I flunked out of my Physics BA after 3 semesters.

Germany's economy alone is 28% of the Eurozone. I don't have to dream.

I really don't see what you're saying. Like 1/2+1/3+1/6 = 1, why can't something similar happen here?

Clearing the denominators and doing some algebra I get x^3+y3+z^3−5xyz-3(x2y+xy^2+x2z+xz^2+y2z+yz^2) = 0 . This doesn't exactly look easier... I tried a parity argument, but it doesn't work. I don't think anything like that can work since multiplied out like this we introduced (0, 0, 0) as a solution and I wrote a python program to brute force this and I found solutions such as (-1, 4, 11) if we allow for negative integers.