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 Nov16 awarded Yearling Nov16 awarded Yearling Nov16 awarded Yearling Jul3 comment $\limsup$ and cluster points In fact, you can show that 1) $\lim \sup x_n$ is a cluster point of $(x_n)$, and that 2) it's greater than any other cluster point of $(x_n)$. That is, $\sup C$ is actually $\max C$. You could try to construct a sequence that converges to $\lim \sup x_n$ (then it's easy to see that $\lim \sup x_n$ is a cluster point of $(x_n)$). Jun24 comment Modular arithmetic for negative numbers @anon I agree, but I understood that the OP was uncomfortable with having a negative right-hand side, and this is a way to avoid that. Jun24 comment Counting number of ways of splitting a card deck Note that sometimes it is acceptable to give an answer in terms of factorials and combinatorial numbers, so you might not actually have to perform the multiplication. Jun24 revised Counting number of ways of splitting a card deck fixed typos, reworded a bit Jun24 suggested approved edit on Counting number of ways of splitting a card deck Jun24 suggested rejected edit on What is a formal definition of series? Jun24 comment Modular arithmetic for negative numbers ... in the equation). Now, to find the values of $m$ that satisfy the congruence, if $k$ is small you could just try some values. Jun24 comment Modular arithmetic for negative numbers With congruences you can always add any multiple of the modulus, since they're all equivalent to $0$. In this case, since $m^2 \equiv -1 \pmod{2k+1}$, and $2k+1 \equiv 0 \pmod{2k+1}$, we have $$m^2 \equiv -1 = -1 + 0 \equiv -1 + 2k+1 = 2k \pmod{2k+1}$$ It's a string of equalities and equivalences, consider each $\equiv$ or $=$, and try to understand what happens in the passage from the left side to the right side of each $\equiv$ or $=$. In any case, this is only one thing you could do (to avoid the negative number; although you should be comfortable with having a negative right-hand side ... Jun24 revised Modular arithmetic for negative numbers latex, capital letter Jun24 comment Modular arithmetic for negative numbers You can note that $m^2 = -1 \equiv (2k+1)-1 = 2k \pmod{2k+1}$. Jun24 suggested approved edit on Modular arithmetic for negative numbers Jun24 revised Eigenvectors of a matrix and its diagonalization fixed typos Jun24 suggested approved edit on Eigenvectors of a matrix and its diagonalization Jun24 revised A proof in number theory dealing with modular congruences. added 122 characters in body Jun24 comment A proof in number theory dealing with modular congruences. @Iyengar Never mind :-) Jun24 comment A proof in number theory dealing with modular congruences. @Iyengar I thought about that, but I'm answering the question that the OP is asking, and since answering a question in the comments is in fact disencouraged, I believe I'm following the guidelines of the site (regardless of the length of the answer and the effort that I put into it). If I'm wrong about this point, I welcome any corrections. Jun24 revised Generating the Sorgenfrey topology by mappings into $\{0,1\}$, and on continuous images of the Sorgenfrey line added latex, fixed formatting