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 Oct15 comment How many 6 digit numbers with 2 or 3 repetitions allowed @calculus sorry for my bad English. I modified the question, maybe this time it is simpler to understand. From 2 repetitions, I tried to mean that every digit (0,..,9) can be used at most 2 times while forming the 6 digit number. Oct15 comment How many 6 digit numbers with 2 or 3 repetitions allowed @CameronBuie yes both of them are true for their cases. sorry for the late reply Apr8 comment Combinatorial explanation to following recurrence relation $a_n = 2 a_{n-1} + a_{n-2}$ @xen I did not say azimut's proof is not combinatorial. I just wanted to say it may not be what Xentius looking for according to his previous questions. But of course only one to confirm this is Xentius. Apr8 comment Combinatorial explanation to following recurrence relation $a_n = 2 a_{n-1} + a_{n-2}$ I guess what @Xentius meant by a combinatorial proof is not this. He seems to be looking for an answer like in this question he asked: math.stackexchange.com/questions/340905/… Mar19 comment Binomial formula for $(x+1)^{1/3}$ (related to Newton's binomial theorem) @Xentius it's called gamma function. Dec16 comment Proof about diameter of a set I guess this should be enough for a) Choose $x_n ∈ A_n$ , and show that the sequence $(x_n)$ satisfies the Cauchy condition. Dec16 comment For which x values, this series is convergent? @DonAntonio Thank you so much. It is a pleasure for me to be a member of the club. = ) Dec16 comment For which x values, this series is convergent? @DonAntonio Ops! you are both right. I edited my answer. Dec5 comment Limit proof of $a^n$ where |a|<1 yeah it is so simple. how could not I think that! :/ I guess I need to take a rest. thank you for your answer. Dec5 comment Limit proof of $a^n$ where |a|<1 @nikita2 you are right. I edited my question. Dec3 comment Boundary and Interior Points of the set: Rational Numbers @TonyK Oh I guess I understood what you wanted to say. In the interval $(X_0-r,X_0+r)$, there are also irrational numbers and therefore it cannot be a subset of $Q$. Right? Nov30 comment Using Euler Phi function to show that there are infinitely many primes I guess I understood why $\phi(m) = 1$. Any selection of (m,n) will not be relatively prime only (m,1) will be relatively prime hence $\phi(m) = 1$. That is brilliant. Thank you so much. Nov30 comment Using Euler Phi function to show that there are infinitely many primes I guess it must go on like this: we know that $\phi(m) = 1$ however according to the multiplicative property of the phi function, $\phi(m) = (p_1-1)*(p_2-1)*...*(p_k-1)$ which is not clearly equal to 1. However I could not exactly understand why $\phi(m) = 1$. Maybe I should give it some time. = ) Nov26 comment Infinitely number of primes in the form 4n+1 proof @GerryMyerson can you show me a way to prove it? Nov26 comment Showing that if a subset of a complete metric space is closed, it is also complete was that all? :D Nov26 comment Showing that if a subset of a complete metric space is closed, it is also complete I am extremely tired so that it has become really hard to think clear, but I added more details about what I have done so far. Could you please take a look and help me to put this proof in a correct shape? Thanks! Nov26 comment Two proofs about metric spaces and one about series @MehenniBenghorbal Oh it just came to my mind: is it OK to directly say that let $\epsilon=\frac{l}{2}$ for a formal proof? I know it does not change anything as long as $\epsilon$>0. But I had a feeling that my professor would not be happy if I directly say let $\epsilon=\frac{l}{2}$. = ) Nov26 comment Two proofs about metric spaces and one about series @AndresCaicedo yes I meant to say "it is close to $l$" but now I am aware that that statement was useless. Nov26 comment Two proofs about metric spaces and one about series @amWhy you are right. I was a little bit panicked to get a fast answer that is why I did such a thing. I have already got answers for the first two questions. So I only deleted the third question and will post it as a separate question. Nov26 comment Two proofs about metric spaces and one about series I am very grateful for your help!