The actual question is as follows:
Suppose you have a sequence of polynomials of degree $n$,
$$P_j(x)=\sum_{k=0}^na_{jk}\,x^k\rightarrow P(x) $$
which is bounded in $n+1$ points for $x$ for all $j$.Prove that the sequence of coefficients $a_{jk}\rightarrow a$ is also bounded . I am supposed to be able to prove it with the Lagrange representation of the polynomial. Yet I don't even have a clue where to begin.
On
Maybe not quite what you are looking for, but here is a more abstract approach which circumvents looking at any formulas.
Define a linear map $Ev$ from the space of polynomials of degree at most $n$ into $\mathbb{R}^{n+1}$, sending $P(x)$ to $(P(x_0),P(x_2),\dots,P(x_n))$ where the $x_i$ are the $n+1$ points you mentioned. Note that this map is invertible since it is between vector spaces of the same dimension and has a trivial kernel as a polynomial of degree at most $n$ can only vanish at $n+1$ points if it is the zero polynomial.
Now the condition that the $P_j(x)$ are bounded at these $n+1$ points means the images of the $P_j(x)$ under $Ev$ are contained inside a bounded region $U$ of $\mathbb{R}^{n+1}$. But then all the $P_j(x)$ are contained in $Ev^{-1}(U)$ which is bounded as it is the image of a bounded region under the linear map $Ev^{-1}$.
Hint: Lagrange polynomials can be used to compute the coefficients of a degree-$n$ polynomial $p(x)$, given $p(x_1), \dots, p(x_{n + 1})$ for distinct values $x_1, \dots, x_{n + 1}$. For your problem, you are effectively trying to bound the coefficients of $p(x)$, given bounds on $p(x_1), \dots, p(x_{n + 1})$. In other words, roughly, you are supposed to show that the Lagrange interpolation procedure is not wildly sensitive to small changes to its input data.