Is it possible to generate a unique real number for each fixed length sequence of real numbers? 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.

You can easily create an injection even for infinite sequences of reals (an injective mapping from sequences to real numbers).  So your request is too weak.
See Boas Primer of Real Functions Exercise 3.13 and its solution in the back.
Also see my thread about this exercise.
