Does y=(−1)xy = (-1)^x where x∈ℝx∈ℝ, change exponentially?

Is y = (-1)^x an exponential curve, or just a sinusoidal one, can it be said to change exponentially as with positive exponents? I’m sure W/A showed this as being sinusoidal with an integer period. But now shows the answer as -1. From the answers I understand that their is no definitive definition but their … Read more

Intuition for interior points and non-open sets

I’m currently reading a book on real analysis. There is a statement right in the introductory section which I’m struggling to intuitively understand. It goes as follows: Assume a sequence (On) of open sets with On:=]−1/n,1/n[. Their countably infinite intersection (infimum) M:=∞⋂n=1On={0} is not an open set because 0 is not an interior point of … Read more

Question about using arbitrary ϵ\epsilon in real analysis proofs

I’ve notice that in a lot of the proofs that are assigned in an undergraduate analysis course, we are often trying to show that some quantity is bounded by an arbitrary epsilon. For example, if I want to show that lim I could try to show that for any \epsilon > 0 I can find … Read more

Proof that the Order topology on R\mathbb{R} has the same basis as the Euclidean topology

I want to prove that the Order topology on R has the same basis as as the Euclidean topology on R. Assume that the only thing we know about the order topology is that it has the open rays as its subbase. My problem is that at some point I have to make some kind … Read more

Fields of intermediate cardinality

Assuming the existence of a cardinal ℵ0<m<2ℵ0, does it follow that there is a subfield of R of cardinality m? Answer Certainly. You can adjoint the subset of cardinality m to Q. First by axiom of choice, you can enumerate the subset of cardinality m, say x0,x1,…,xm. Then construct the tower of extension Q→Q(x0)→⋯. Let … Read more

What is the rigorous definition of the ratio of two numbers?

I have tried learning about ratios and proportions from a couple of books, but I have problems with the approaches they take. First of all, a definition is never actually given. Examples are given concerning only natural numbers. For example one definition I encountered was the ratio of object x to object y is m:n … Read more

Does the set of vectors S={v1,v2,0}S =\{v_1,v_2,{0}\} span the real 33-dimension?

Im trying to prove that a set of vectors, specifically in 3D, spans the real third dimension. In general, i have read that if a set of vectors contains 3 non-coplanar vectors, they would span the real 3D. But I’m confused as to whether this applies to a set containing the zero vector as well? … Read more

How to prove that infinity (∀n∈N:x>n\forall n \in \mathbb N: x>n) is not a real number?

By infinity, I mean a number x, such that x>n for any natural number n. Here, x+1 would just be x+1, so we couldn’t use something such as x+1=x>x to show that it doesn’t exist. Its multiplicative inverse, ϵ, would also be a real number, smaller than any other positive real number. My first thought … Read more

Generalised Triangle Inequality

For x,y,z∈R and usual absolute value, we have the following form of the triangle inequality: |x+y|≥|x|−|y| What I want is a generalised form like: |x+y1+⋯+yn|≥|x|−n∑i=1|yi| But I don’t know for sure if it’s true, and I’m stuck with proving it. I tried generalising the first formula for y=y1+⋯+yn but getting nowhere. Any help would be … Read more

Why real numbers doesn’t end with infinitely many 9s?

In the text TAOCP A real number is a quantity x that has a decimal expansion x=n+0.d1d2d3… where n is an integer, each di is a digit between 0 and 9, and the sequence of digits doesn’t end with infinitely many 9s Why is such a case ? Isn’t all the numbers between 0 and … Read more