Prove that if $f_n : [0, 1] \to \mathbb{R}$ is a sequence of polynomials of the same degree and $f_n$ converges uniformly to $f$ then $f$ is a polynomial.
I've seen similar questions on MSE, but with unbounded domain and without assumption that degrees of $f_n$'s are the same.
Let $f_n(x)=a_d(n)x^d+a_{d-1}(n)x^{d-1}+\dots+ a_0(n)$ where $d$ is the common degree. We have that:
1) Since for $k=0,1,\dots,d$, $$f_n(k/d)=a_d(n)(k/d)^d+a_{d-1}(n)(k/d)^{d-1}+\dots+ a_0(n)\to f(k/d)$$ it follows that the sequence $(a_k(n))_n$ converges to some $A_k\in\mathbb{R}$ for any $k=0,1,\dots, d$ (use the Vandermonde matrix with the powers of $k/d$).
2) Let $P(x)=A_d x^d+A_{d-1}x^{d-1}+\dots+ A_0$, then, as $n\to +\infty$, $$\max_{x\in [0,1]}|P(x)-f_n(x)|\leq \sum_{k=0}^d|A_k-a_k(n)|\to 0$$ and therefore the limit function $f$ is the polynomial $P$.