Proving linear independence of infinite set (monomials) I would like to prove that the set of monomials is linearly independent in the complex linear space $C(\mathbb R)$. I understand the definition of linear independence and I'm stuck on how to prove linear independence for all subsets. Where would one start with such a proof?
 A: One can prove this by showing the following statement for all $n$:

The functions represented by $1,\,x,\,x^2,\ldots,\,x^n$ are linearly independent.

Note that, if adding a single element to a linearly independent set creates a linearly dependent set, then that new element must be writable as a linear combination of those initially in the set. Thus, all we must prove is:

The function $x^n$ cannot be written as a linear combination of $1,\,x,\,x^2,\ldots,\,x^{n-1}$.

But this isn't so hard: Note that $\lim_{x\rightarrow\infty}\frac{x^m}{x^{n}}=0$ for $m<n$. We can therefore deduce that, for any $P(x)$ writable as a linear combination of such $x^m$ we have $\lim_{x\rightarrow\infty}\frac{P(x)}{x^n}=0$ which implies $P(x)$ is not $x^n$, completing the proof. The linear independence of the whole set follows from the independence of these subsets, as any linear dependence among the whole set could only involve finitely many monomial.
A: A poly nomial is, by definition, a finite linear combination of monomials (i.e., of the form $\sum_{i=1}^n a_i m_i$ with the $a_i$ elements of the underlying field and the $m_i$ (distinct) monomials). A polynomial is $0$ if and only if all its coefficients are $0$ and that's exactly the claim that the set of monomials is linearly independent.
