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Problem:

If a function $g:\mathbb{R}\rightarrow \mathbb{R}$ which is continuous, can be uniformly approximated by polynomials on the real numbers $\mathbb{R}$, then it is required to prove that this function can be nothing but a polynomial.

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If $\{P_n\}$ is a sequence of polynomials which converges uniformly to $g$ on the real line, then $\{P_{n+1}-P_n\}$ is a sequence of polynomials which converges uniformly to $0$ on the real line. So for $n$ large enough, says larger than $n_0$, $P_{n+1}-P_n$ is bounded on the real line, therefore equal to a constant says, $c_n$. Hence $P_n=\sum_{k=n_0}^{n-1}c_j+P_{n_0}$. Since $\{P_n \}$ is Cauchy, the series $\sum_{k\geq n_0}c_k$ is convergent. Denote $c$ the limit; we get that $g=c+P_{n_0}$, a polynomial.

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    I realize I didn't defined $n_0$, now it's done.2012-04-25