Recent content by R_beta.v3

Sorry. I just want to make sure that my proof is correct? I'm studying by myself.

Homework Statement I'm trying to prove these two theorems a) if ## 0 \leq f(x) \leq g(x) ## for all x ## \geq 0 ## and ## \int_0^\infty g ## converges, then ## \int_0^\infty f ## converges b) if ## \int_0^\infty |f| ## converges then ## \int_0^\infty f ## converges. Obviously assuming...

Well ##B## is bounded below by ##\alpha##, so ##\alpha## is a lower bound of B, and since ##\beta## is the greatest lower bound of ##B##, ##\alpha \le \beta##

Homework Statement If ##f## is continuous on ##[a,b]## and ##f(a) < f(b)##. Prove that there are numbers ##c, d## with ##a \le c < d \le b## such that ##f(c) = f(a)## and ##f(d) = f(b)## and if ##x \in (c,d)## then ##f(a) < f(x) < f(b)##. Homework Equations The Attempt at a...

Correction That should do. Sorry for the errors, typing math symbols is not as simple nor fun as using a pen.

Oh, yes, sorry, I am too used to using the x as the value approaching a, what I actually meant is the limit as {y approaches x} of f(y).

Homework Statement This problem took me a lot of time if g(x) = \lim_{y\rightarrow x} {f(x)} exist for any x, then g is continuous. Homework Equations The Attempt at a Solution \lim_{x\rightarrow a^+} {f(x)} = g(a) , so if ##\epsilon > 0 ## then there is an ##\delta_1 > 0## such that...

Really great advices, thanks everyone. The Raspberry Pi looks great. I am sure I'll buy it soon.

I really want to be a programmer. But I am more interested in operating systems(I always get goosebumps thinking about coding an operating system), programming languages, compilers, emulation, AI, embedded systems, assembly, parsing, hardware, etc. Where do you even begin for that stuff...

My favorite science fiction shows are, in order, Babylon 5, Farscape and Stargate SG1. I tried watching Sanctuary and Falling Skies but I didn't enjoy them. Continuum is fun. In my opinion, if there is one sci-fi series you have to watch, it should be Babylon 5.

Of course! I can't believe that I didn't do that. Thanks.

Well, nothing. I used -1 and 1 because they are simple. The "proof" is basically the same with other numbers.

Homework Statement if ##f## is a continuous function, ##f(x) > 0## for all x, and \lim_{x\rightarrow +\infty} {f(x)} = 0 = \lim_{x\rightarrow -\infty} {f(x)} Prove: There is a y such that ##f(x) \leq f(y)## for all x I'm assuming that the domain of f is ℝ. Homework Equations The...

I haven't read many science-fiction books, but recently I began reading Asimov books. -The End of Eternity -Prelude to Foundation -Forward the Foundation I liked them a lot. Especially "The End of Eternity".

Homework Statement If f is continuous, and f(x) = 0 for all x in A, where A is a dense set. Then f(x) = 0 for all x. I am using the following definitions: A set of real numbers A is dense if every open interval contains a point of A. And the limit definition for a continuous function...