Recent content by laughingebony

You may not find this satisfactory, but once the dot product is defined, the angle between two vectors a and b is usually defined to be arccos(\frac{a \bullet b}{\left| a \right| \left| b \right|}), which should seem reasonable from the fact that this can be proved geometrically in two and three...

In fact, the integral of an integrable function f over [a,b] is often written \int^{b}_{a}f. Notice that even in the English description, "the integral of f over [a,b]," I didn't even refer to any variables. Likewise, if the the second endpoint of the interval is a variable, we can write the...

To understand the Second Fundamental Theorem intuitively, let x_{1} and x_{2} be points close together in [a,b] with x_{1}<x_{2}. Define \Delta x = x_{2}-x_{1} and \Delta A to be the area under the graph of f from x_{1} to x_{2}. (Assume f is nonnegative on [a,b].) Now pick a point c between...

M1 is actually both bounded and convex. I think you would benefit from seeing a specific example of how such properties can be proved formally, so I'll prove this claim. I should also mention that I know very little topology, so I can't answer you question about what topologists do in...

It depends on whether the infinite limits have the same sign. If, say, \lim_{x\to a} f(x) = \infty while \lim_{x\to a} g(x) = -\infty, then we have to use other methods to investigate the limit. But, informally speaking, if \lim_{x\to a} f(x) = \lim_{x\to a} g(x) = \infty (note: same sign)...

Regarding the first question: If I gave you 10 apples and asked you to divide them into 2 groups of 5 apples each, would you put any of the apples in both groups simultaneously? Regarding the second question: Even though the author says there's no A team and no B team, I think it...

Not really, because that already assumes that irrational numbers exist. Their existence is a consequence of how the system of real numbers is built up from the system of rational numbers (which, as has already been mentioned, is done via Cauchy sequences or Dedekind cuts). Part of the reason...

It means precisely that for each positive number ε, there exists a natural number K such that for each natural number k≥K, \left|\left(\sum^{k}_{n=1}\frac{1}{2^{n}}\right)-1\right|<\epsilon (I changed the starting index to 1, which is more in line with Zeno's paradox.) If you want to interpret...

Convergence of a series has a specific mathematical definition. The fact that, say, \sum^{\infty}_{n=0}\frac{1}{2^{n}} (the series relevant to Zeno's paradox) converges means only that we have a reasonable way of dealing with this series mathematically. It does not imply, for instance, that...

That's not standard. I don't have access to a copy of Rudin at the moment, but I suspect you've misread something. {1,2,3,...} (i.e., the set of natural numbers) is the domain of the sequence. The range is {an|n is a natural number} (i.e., the set of sequence values). With the correct...

One way to define the natural logarithm is to start with the differential equation f'(x)=1/x, subject to f(1)=0, and show that there is a unique function defined on the interval (0,∞) that satisfies those conditions. We then call that function the natural logarithm. If you don't know any...

Can you prove this? If you can, see if you can use the same technique to prove that β falls between N+(m/n) and N+[(m+1)/n], for some m.