In vector space, let $\,x_k=\left(a_k,b_k,c_k,d_k,\ldots\right)\,$ be a Cauchy sequence. When I prove $\mathbb{R}^n$ is complete, can I claim that each element in $\,x_k\,$ is Cauchy in $\,\mathbb{R}$, and $\,\mathbb{R}\,$ is complete. Thus, there exists a limit for each element in $\,\mathbb{R}.\,$ Then, let $\,x = \left(a,b,c,d,\ldots\right)\,$ be a limit point for the Cauchy sequence, $\,x_k\,$ and hence $\,x \in \mathbb{R}^n$, $\,\mathbb{R}^n$ is complete.
Am I making a valid argument?
The idea is good, but you are using improper language:
you should write the sequence as $\{ x_k \}_k = \{ (x^1_k, \dots , x^n_k) \}_k$ (traditionally $(a,b,c, \dots )$ means that this goes on to infinity, while this is not the case). The numbers $x^i_k$ are not the elements of $x_k$: their correct name is "components". For example, $1$ and $3$ are the components of the vector $(1,3)$.
you should somehow prove that for all $i$ you have $\{ x^i_k \}_k$ is a Cauchy sequence converging to some $x^i$, and call $x=(x^1, \dots , x^n)$.
you should somehow prove that $x_k$ converges to $x$.
The second and third steps need to work with some metric on $\Bbb{R}^n$, otherwise it is meaningless to say that something is a Cauchy sequence.