On the associative property of a binary operation of the fundamental group.

I was reading about the proof of associativity property of the operation on the fundamental group here. The book gives the following diagram then it says the reader should supply the elementary geometry necessary to derive the homotopy But I don’t know how to do that. Please show me how to derive the above homotopy … Read more

Help me understand this sequence problem

Today, I encountered a problem in “Problem-Solving Strategies” by Arthur Engel (Chapter $9$. Sequences, page-$225$): Prove that there does not exist a monotonically increasing sequence of nonnegative integers $a_1,a_2,a_3,…$ so that $a_{nm}=a_n+a_m$ for all $n,m \in \mathbb{N}$. I could not prove it. So, I checked the solution provided in the book. It was: Solution: For … Read more

Suppose RR is a partial order on AA and B⊆AB \subseteq A. Prove that R∩(B×B)R \cap (B \times B) a partial order on BB.

Can somebody show me how to prove this? I would much appreciate it if one could show the givens and goals similar to how it is set out in Velleman’s ‘how to prove it’ book, though any help would be good. Thanks. Answer I don’t know Velleman’s book, but I support your seek for guidance. … Read more

Let ¨x=2x\ddot x=2x. If ˙x\dot x is 00 when x=1x=1. Find ˙x(x=3)\dot x(x=3).

I was given this problem in a physics class, and below is the answer by the professor. (x is position, v=˙x is velocity, a=˙v=¨x acceleration). Let a=2x. If v is 0 when x=1. Find v when x=3. Professor’s answer: We can rewrite a=dvdt=dvdx⋅dxdt=dvdxv(t), thus dvdxv(t)=2x⟺vdv=2xdx⟺∫v0vdv=2∫31xdx⟺v2(3)2=8∴ I belive this answer is an atrocity: He allegedly arrived … Read more

Proving that 14(5)+15(6)+16(7)+⋯+1(n+3)(n+4)=n4(n+4)\frac{1}{4(5)}+\frac{1}{5(6)}+\frac{1}{6(7)}+\cdots+\frac{1}{(n+3)(n+4)}=\frac{n}{4(n+4)} by induction

I’ve proved the base case where n=1 and made the assumption that n=k is true, but I’m stuck on the n=k+1 part. I just cannot seem to get the algebra to work in my favor. Here is the original: 14(5)+15(6)+16(7)+…+1(n+3)(n+4)=n4(n+4)∀n∈N I get it to the following form and just run in circles: 14(5)+15(6)+16(7)+…+1(k+3)(k+4)+1(k+4)(k+5)=k+14(k+1+4) Simplifying k4(k+4)+1(k+4)(k+5)=k+14(k+1+4) … Read more

Proving by induction that \sum\limits_{k=2}^n \frac1{k^2}\le 1\sum\limits_{k=2}^n \frac1{k^2}\le 1

I’m having trouble proving this by induction. We need to show that P(k+1) is true: \sum_{i=2}^{k+1} \frac{1}{i^2}\leq 1. Don’t know where to go from here. Any help? Answer Hint. Try to apply induction to \sum_{k=2}^n \frac1{k^2}\le 1-\frac1{n},\qquad n\ge2. AttributionSource : Link , Question Author : Chance Gordon , Answer Author : Olivier Oloa