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mahler1
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Homework Statement .
Let ##A## be the set of sequences ##\{a_n\}_{n \in \mathbb N}##:
1) ##a_n \in \mathbb N##
2) ##a_n<a_{n+1}##
3) ##\lim_{n \to \infty} \frac {\sharp\{j: a_j \leq n\}} {n}## exists.Call that limit ##\delta (a_n)## and define the distance (I've already proved this is a distance) ##d(a,b)=|\delta (a) -\delta (b)|+k^{-1}## where ##k=\{min j : a_j≠b_j\}## and ##d(a,a)=0## for any two sequences ##a## and ##b##. Prove that the function ##g:(A,d) \to [0,1]## defined as ##g(\{a_n\})=\delta(\{a_n\})## is surjective and continuous. The attempt at a solution.
I didn't have problems to prove that this function is continuous (in fact, it's uniformly continuous), but I am totally lost at the surjectivity part. The only elements in the codomain to which I could associate two members of the domain where ##0## and ##1##. Someone suggested me to consider for each ##x \in (0,1)##, the sequence ##{x_n}## with ##x_n=\lfloor \frac {n} {x}\rfloor##. It's immediate this sequence is of natural numbers, but how can I prove that ##\lfloor \frac {n} {x}\rfloor <\lfloor \frac {n+1} {x}\rfloor ## and that ##\delta (x_n)=x##?
If someone has a better suggestion/idea of how could I prove surjectivity, you are welcome to tell me.
Let ##A## be the set of sequences ##\{a_n\}_{n \in \mathbb N}##:
1) ##a_n \in \mathbb N##
2) ##a_n<a_{n+1}##
3) ##\lim_{n \to \infty} \frac {\sharp\{j: a_j \leq n\}} {n}## exists.Call that limit ##\delta (a_n)## and define the distance (I've already proved this is a distance) ##d(a,b)=|\delta (a) -\delta (b)|+k^{-1}## where ##k=\{min j : a_j≠b_j\}## and ##d(a,a)=0## for any two sequences ##a## and ##b##. Prove that the function ##g:(A,d) \to [0,1]## defined as ##g(\{a_n\})=\delta(\{a_n\})## is surjective and continuous. The attempt at a solution.
I didn't have problems to prove that this function is continuous (in fact, it's uniformly continuous), but I am totally lost at the surjectivity part. The only elements in the codomain to which I could associate two members of the domain where ##0## and ##1##. Someone suggested me to consider for each ##x \in (0,1)##, the sequence ##{x_n}## with ##x_n=\lfloor \frac {n} {x}\rfloor##. It's immediate this sequence is of natural numbers, but how can I prove that ##\lfloor \frac {n} {x}\rfloor <\lfloor \frac {n+1} {x}\rfloor ## and that ##\delta (x_n)=x##?
If someone has a better suggestion/idea of how could I prove surjectivity, you are welcome to tell me.
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