Accumulation point(s) was one of the topics in our math analysis lecture today. I'm not familiar with the term of a topological space so I would like to find the solution to a problem task using the tools we were introduced with in the lectures. We are supposed to find,if possible, a sequence whose accumulation points are all the integers, in other words: $$\exists? \;(a_n)_n\;\;\;\text{such that}\;\;\mathbb Z\;\;\text{is the set of all its accumulation points}?$$ I was thinking of the floor & ceiling functions, but the problem is that $$|\mathbb Z|<|\mathbb R|$$ $$\text{if}\;f(x)=\lfloor x\rfloor\;\text{or} \;g(x)=\lceil{x}\rceil$$ $$\nexists\;f^{-1}(x)\;\wedge\;\nexists\;g^{-1}(x)$$ I need an image of an injective function. We proved the Weierstrass theorem and used Cantor's theorem several times, but this sequence, if exists, is divergent. Another question our professor proposed is what an accumulation point can be anyway? And a similar question:
$$\exists? \;(a_n)_n\;\;\;\text{such that}\;\;\mathbb R\setminus\{0\}\;\;\text{is the set of all its accumulation points}?$$ I had sets of linear combinations of exponential functions in my mind, but I don't know how to express that without mixing limits and accumulation points.