How can I prove that?I tried to find a solution but I don't know how to do it.
123164
My functional analysis textbook says   "The metric space $l^\infty$ is not separable."   The metric defined between two sequences $\{a_1,a_2,a_3\dots\}$ and $\{b_1,b_2,b_3,\dots\}$ is $\sup\limits_{i\in\Bbb{N}}|{a_i-b_i}|$.   How can this be? Isn't the set of sequences containing complex numbers with rational coefficients the required countable dense subset of $l^\infty$?  Thanks in advance!
74091
Problem: Show that if $(b,c)=1$, then for any integer $a$ we have $(a,bc)=(a,b)(a,c)$. [Hint: Prove that each member of the alleged equation divides the other.]  This is the way I proved it (without the hint): $(a,b)(a,c)=(a(a,b),c(a,b))=(a^2,ab,ac,bc)=(a^2,bc,a(b,c))=(a^2,a,bc)=((a^2,a),bc)=(a,bc)$  But how should I prove it using that hint or at least a similar proof.
94331
Problem: Hi, I'm trying to solve the following proof:  If $a &gt; b \geq 0$, Prove that $a^2 &gt; b^2$.  But I'm a bit new with proofs and I don't manage to write a proper one. I would be very grateful if someone could explain me which are the steps required to demonstrate the proof. Thank you.
85649
I can prove the converse of it, but I cannot do this one. Here is the problem:     Prove that the implicit function theorem implies the inverse function theorem.
26450
I managed to show that the limit exists, but I don't know how to compute it.  EDIT:  There are initial terms: $a_1=1$ and $a_2=2$.
122604
How can I prove      $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$$   I don't know which method can be used for this?
42725
I can intuitively see that this is true, but I'm having a very hard time proving it. I'm actually not even quite sure where to begin. I tried using the inequalities that define the floor function, and I'm pretty sure this is the way to go, but I'm failing to make any kind of progress.  I appreciate any help.
83554
I'm really struggling with "simple" proofs, and I could only come up with this. I googled for the solution and know how to do it now, but I first want to know if there's anything wrong with my solution? $$ (AB)^{-1}=B^{-1}*A^{-1}. $$ For this equation to hold, $B$ must be invertible and so must $A$. //
123354
I have to prove that. While I know this is true by thinking about it I'm having a lot of trouble actually writing the proof
54080
I do not know an example. Will ask question if in doubt of the proofs provided thank you!!
103821
I saw this problem recently and found an elegant solution to it, and was curious to see if anybody would think of something else. Nice solutions to nice problems are fun to see!  Problem: Prove that, for all non-negative integers $k$, there exist integers $a$, $b$ such that $a^2+b^2=5^k$. Bonus: Prove that there exist such $a$ and $b$ that furthermore satisfy $(a,b)=1$.
107054
We are considering that $h &gt; 0$. I just don't really know how to approach this question; it seems that by taking the limit of the inequality as h goes to 0, each part of the inequality is going to be equal to $f'$, but I don't really understand why the fact that $f'$ is increasing is relevant to finding a solution to this question (and how to answer the question based on that constraint).
12307
I've tried deriving a proof for this using the handshaking lemma, but that was unsuccessful. Could someone please explain to me how they would do this proof?
7044
I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :)
9732
It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this?
8161
I posted a question somewhere that said...     Can anyone tell me how I can solve this?   ...but someone edited it to...      Can anyone tell me how can I solve this?    ...and it was accepted.  That's wrong isn't it? Can someone explain how that's wrong?
82274
I am not sure how to show the second part. Isn’t it obvious from the fact that    G and H are infinite?