Let A be the set of all sequences of real numbers of size $n$. Does there exist an injection from A to R?
I know this is possible if we are only considering integers instead of real numbers; But I am not sure if it is possible if we consider real numbers instead.
For integers, we can generate a unique integer using the following method: Let S be a sequence of integers of size n. $S = s_1,s_2,\ldots,s_n$. Let $P = p_1,p_2,\ldots,p_n$ be the sequence of $n$ primes. Then $f(S) = (p_1^{s_1})(p_2^{s_2})\cdots(p_n^{s_n})$ creates a unique integer for each sequence $S$.
If each $s_i$ was a real number instead, would $f(S)$ still be an injection? If not, is there an alternative invective function from A to R?
edit:
I fixed some of my poor wording.
I am trying to find an injection function from A to R. Such a function does exist and the function I proposed clearly does not work (From the comments).
If possible, I would like to find an injective function that does not involve directly manipulating the decimal expansions.