Recent content by georgetown13

Homework Statement Determine the limit and then prove your claim. limx\rightarrow\infty (1+\frac{1}{x^2} }) xHomework Equations I know that the formal definition that I need to use to prove the limit is: {limx\rightarrow\infty (1+\frac{1}{x^2})x=1}={\forall \epsilon>0, \exists N > 0, \ni x>N...

Do I have to use induction for this proof?

Homework Statement Let U be a countable set. For j=1, 2, 3, ... define Fj= {(x1, x2, ... , xj): xi \in U for all i=1,2,...,j} Prove that j=1,2,3,...,Fj is countable. Homework Equations The Attempt at a Solution I'm having trouble understanding what the set Fj contains. F1={x1} F2={x1, x2}...

Yes, but I can't just provide an example to prove the statement, right? I understand the general concepts behind this proof but am having a difficult time putting it down in mathematical terms.

Homework Statement Let U and V both have the same cardinality as R (the real numbers). Show that U\cupV also has the same cardinality as R. Homework Equations The Attempt at a Solution Because U and V both have the same cardinality as R, I that that this means \exists f: R\rightarrowU that is...

Let S be the subspace of R^3 spanned by x=(1,-1,1)^T. Find a basis for the orthogonal complement of S. I don't even know where to start... I would appreciate your help!

Homework Statement Let x1, x2, and x3 be vectors in R^3. If x1 is orthogonal to x2 and x2 is orthogonal to x3, is it necessarily true that x1 is orthogonal to x3? Homework Equations I know that if x1 is orthogonal to x2 and x2 is orthogonal to x3, then... (x1)^T*x2=0 (x2)^T*x3=0...