Recent content by Mixer

Thank you for your reply! So my proof is not perfect as I expected. ;-) The case where only three points are collinear seems to be difficult. If points P1, P2 and P3 are collinear then we can define a line that passes trough P1 and P4 or P1 and P5. None of these lines are same as the line...

I am just trying prove this by contradiction. So I am just assuming that point P5 is a separate point. It could be any other point of cource. From there I can come to conclusion that if the points are not collinear then it is possible to define a line that passes trough exactly two points...

Homework Statement There are five points on a plane. There is no line that passes trough exactly two points. Prove that five points are collinear. Homework Equations - The Attempt at a Solution I am just trying to confirm that my proof is correct. I am trying to prove this by contradiction...

Greetings, Found this grid from Internet. There was no hint if this was a puzzle but I pretty much sure that it is. Has someone seen this kind of grid before? If so, how to solve this? The link to the grid is: https://www.geocaching.com/geocache/GC6FQYC_pvm-6-chimney

Thank you for reply! So are you saying that I should take g(t) = t0 + 1 for all t. Then \left\|f - g\right\| = sup_{t \in [a,b]} |f(t) - g(t)| \geq |f(t_0) - g(t)| = |0 - t_0 -1| = |t_0 + 1| Therefore set S is not dense in C[a,b] ?

Homework Statement Let [a,b] \subset \mathbb{R} be a compact interval and t0 \in [a,b] fixed. Show that the set S = {f \in C[a,b] | f(t_0) = 0} is not dense in the space C[a,b] (with the sup-norm). Homework Equations Dense set: http://en.wikipedia.org/wiki/Dense_set sup -...

I think I got it now. So let's assume that there exist no such r. Therefore there is sequence (x_n) \subset \mathbb{R^n}\setminus U such that \|{x_n - k_{n_j}}\| < \frac{\epsilon}{2}. Now k_{n_j} is a converging subsequence in K. Because K is compact there exists subsequence k_{n_j} which...

Well, only difficulty I see in the proof is that how to show the corresponding subsequence of points not in U also converges to k. Of cource if points get arbitarily close, so the distance between points is less than r > 0.. ?

Oh, I think I found it. Page 2 of this pdf section (d) : http://math.ucr.edu/~res/math205A/math205Asolutions3.pdf . Am I done? :D

I feel I'm completely lost now.. Of cource since K is compact then it is closed and bounded. Because it is bounded then for every u \in K \|u\| \leq M. I really need some hint now..

Erh... Ok, so my reasoning does not work. How about the fact that since U is open, then every element x \in U has an open ball-neigbourhood B_r{x} \subset U. Then every element in K has that neigbourhood also. Take all these neighbourhoods and then this is an open cover of K. Because K is...

Homework Statement Let K \subset \mathbb{R^n} be compact and U an open subset containing K. Verify that there exists r > 0 such that B_r{u} \subset U for all u \in K . Homework Equations Every open cover of compact set has finite subcover. The Attempt at a Solution I tried...

Ok, thanks I'll show the solution I got in first post next lecture. After all, that is the only reasonable solution I could come up. If it is wrong, then it is. Thank you guys!

No, my teacher won't show any examples during lecturers and yes I do attend classes. This is not an online cource. I tried to find older versions of the lecture notes, but I failed :( But basically I know lens formula which is mentioned in first post and I know what diopter is. I also know...

My cource material has no examples :( Here is the cource material: https://moodle2.tut.fi/pluginfile.php/116692/mod_resource/content/0/Chapter5.pdf I may be wrong, but if the objects appear farther away than they are and velocity is distance divided by time would it make velocity to seem...