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.
 A: Pick a bijection from $\mathbb{R}$ to the set of all infinite sequences of, say, non-negative integers (exercise if you don't already know that this is possible; you can build such a bijection out of continued fraction exansions). Then an element of $\mathbb{R}^n$ (the set of ordered $n$-tuples of real numbers) can be identified with an ordered $n$-tuple of infinite sequences of non-negative integers $a_{i, j}, 0 \le i \le n-1, j \in \mathbb{N}$. Now one can simply interweave these sequences to get a single infinite sequence by setting
$$b_{jn + i} = a_{i, j}$$
and this is clearly a bijection. 
A: I wasn't sure whether we were considering all finite sequences of reals, or just sequences of a predetermined length, so I answered for all finite sequences. Of course, this also works for a predetermined length sequence.
Suppose we have a sequence $\{x_k\}_{k=1}^n$ of real numbers. Generate an integer based on the signs of those reals as follows:
$$
\sigma(\{x_k\})=2^{n-1}-1+\sum_{k=1}^{n-1}2^{k-1}\frac{1+\rm{sign}(x_k)}{2}
$$
taking $\rm{sign}(0)=1$. Interlace the digits of the absolute values of the $x_k$ (not ending in repeating $9$s) to get a unique non-negative real number $\mathscr{I}(\{x_k\})$ and map it to $[0,1)$ with
$$
\phi(x)=\frac{x}{\sqrt{x^2+1}}
$$
Then we get a unique real using
$$
\rm{sign}(x_n)\left(\phi(\mathscr{I}(\{x_k\}))+\sigma(\{x_k\})\right)
$$

Simpler method:
Get a unique real using the seqence of positive reals $\left\{e^{x_k}\right\}$ with
$$
n+\phi\left(\mathscr{I}\left(\left\{e^{x_k}\right\}\right)\right)
$$
A: 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.
