Let $f:[a,b] \rightarrow \mathbb{R}$ be of class $C^n$. Let $ x\_0, ..., x\_m$ be different numbers from $[a,b]$. 


Does for each $\varepsilon >0$ there exist a polynom $P$ such that $P^{(k)}(x\_i)=f^{(k)}(x\_i)$ for $i=0,...,m$, $k=0,...,n$ 
and $sup\_{x \in [a,b]} |f(x)-P(x)|< \varepsilon$?



Approximation by polynomials
The problem may be split into two independent and classical ones: the Hermite interpolation, and the Weierstrass approximation. 


First, we want a polynomial $p\in \mathbb{R}[x]$ with given derivatives at some given nodes $x \_ 0,\dots, x \_ m $. This is an instance of the [Hermite interpolation problem](http://en.wikipedia.org/wiki/Hermite_interpolation); yours has exactly one solution $p$ with $ \operatorname{deg}(p) < (m+1)(n+1) $ (the degree one would expect in terms of number of linear conditions).
So, given your $f\in C^n$, you can find a polynomial with $p^{(j)}(x \_ i)=f^{(j)}(x \_ i)$ for all $0 \le i \le m$ and $0 \le j \le n$. 


Second, as a consequence,  

$$\frac{f(x)-p(x)}{\prod \_ {i=0}^ m(x- x \_ i)^n}$$
is (extends to) a continuous function on $[a,b]$ that vanishes on the points $x \_ i$. By the Stone-Weierstrass approximation theorem there is a polynomial vanishing on the points $x \_ i$ as well, whose uniform distance from that function on the interval $[a,b]$ is less than, say, $\epsilon (b-a)^{-n(m+1)}$. In other words, there is a polynomial $q\in \mathbb{R}[x]$ such that


$$\bigg\| \frac{f(x)-p(x)}{\prod \_ {i=0}^ m(x- x \_ i)^n} - q(x) \prod \_ {i=0} ^ m(x- x \_ i) \bigg\|\_{\infty, [a,b]} < \epsilon (b-a)^{-n(m+1)}\ , $$


therefore the polynomial $P(x):= p(x)+ q(x) \prod \_ {i=0} ^ m(x- x \_ i) ^{n+1}$ fullfills the requirements, for $P^{(j)}(x \_ i)=p^{(j)}(x \_ i)=f^{(j)}(x \_ i)$ for all $0 \le i \le m$ and $0 \le j \le n$, and 
$$\|f- P\|\_{\infty,[a,b]} < \epsilon\ .$$


**btw.** Incidentally, some time ago I happen to notice that one can find the solution of the Hermite interpolation problem as an application of the Chinese Remainder Theorem in the ring of polynomials, and wrote [here](http://en.wikipedia.org/wiki/Chinese_remainder_theorem#Applications) the details. 


**edit.** As to why The set $A$ of all polynomial functions on $[a,b]$ that vanish at given points $x\_0,\dots, x\_m$ is dense in all continuous functions on $[a,b]$ that vanish in $x\_0,\dots, x\_r$. One way, a bit abstract but quite immediate is, to see it as a corollary of the Stone-Weierstrass theorem (A separating closed algebra of real valued functions on a compact space $X$ is either $C(X)$ or a maximal ideal $M\_x\subset C(X)$, the set of all functions vanishing at $x$). Consider $X=$ the topological quotient of $[a,b]$ obtained identifying all points $x\_i$ to a point $\xi$. All functions in $A$ factor through to the quotient map, and define a closed separating algebra of continuous functions on $X$ that vanish on the identified point $\xi$. Thus, this algebra contains all continuous functions on $X$ that vanish on $\xi$, which is the thesis read on the quotient. 


Note that the same construction holds in general, and provides a characterization of all closed algebras $A$ of continuous functions on a compact space $X$: identifying all points that are not distinguished by the functions of $A$ (that is, under the equivalence relation $x R\_A y$ iff $f(x)=f(y)$ for all $f\in A$) one gets a Hausdorff compact quotient space (whether or not $X$ is Hausdorff), and the quotient map $\pi: X\to X/{R \_ A}$ induces an isometric isomorphism of algebras $f\mapsto f\circ \pi$ of either $C(X/{R\_A})$ or a maximal ideal of it onto $A$; conversely, any Hausdorff quotient of $X$ produces a closed sub-algebra of $C(X)$ this way.


Another way to see it is as a corollary of the classic Weierstrass theorem: Consider $P$ as in your comment below, then add a perturbation $L$ that makes $P+L$ vanish on the points $\{x\_i\}$; this has been clearly explained in Ilya Bogdanov's answer. Here you don't have derivatives and $L$ is just a Lagrange interpolation polynomial, which is small in the uniform norm because it is small on the points $\{x\_i\}$. 


