/sci/ - Science & Math

/sci/ approved movies

>yes, anon, we already know interstellar

>>10067855
Let x(n) be a sequence in [a, b] which converges to x(0) (x(0) being an element of [a, b]). We must prove that f(x(n)) converges to f(x(0)).

Given ε > 0, consider the inequality |f(x(n)) - f(x(0))| < ε. This means f(x(0)) - ε < f(x(n)) < f(x(0)) + ε.

For ε small enough, we can assume that both f(x(0)) - ε and f(x(0)) + ε belong to [a, b]. Since f has the Intermediate Value Property, there exist two real numbers v and w in [a, b] such that:

f(v) = f(x(0)) - ε
f(w) = f(x(0)) + ε.

We know that f is monotonic; let's assume it is increasing (the argument is analogous for decreasing functions). Then, since f(v) < f(x(0)) < f(w), it follows that v < x(0) < w. Since x(n) converges to x(0), for large enough n we always have v < x(n) < w; and from this it follows that for large enough n we always have f(v) < f(x(n)) < f(w); in other words we have f(x(0)) - ε < f(x(n)) < f(x(0)) + ε (because of the definition of f(v) and f(w)). This proves that f(x(n)) converges to f(x(0)) and therefore that f is a continous function.

>>10037387
>>10037395
agree
>>10037401
created

https://en.wikipedia.org/wiki/Foundations_of_mathematics#Formalism
You should be able to grasp this.

>>10037411
>Could math exist as qualities of things?
it's qualities, objects, actions, and relationships
there isn't just one things, but a thing called one as well