# Upper bound of a uniformly converging sequence of polynomials

Let $$k\geq 2$$, and let $$P_k$$ be a sequence of polynomials, such that:

1. $$P_k=\sum_{n=2}^{k+1}a_{n,k}X^n \in \mathbb{Q}[X]$$, $$a_{2,k}\neq 0$$, $$\deg P_k \leq k+1$$, and consider $$P_k :[0,1]\rightarrow \mathbb{R}$$ as a real valued function.
2. $$P_k(1)=\frac{1}{k(k+1)}$$ and $$\mid a_{n,k}\mid < \frac{2}{k}$$, for all $$n$$ and $$k$$.
3. for any $$a<1$$, $$\max_{x\in[0,a]} \mid P_k(x)\mid \rightarrow 0$$, as $$k\rightarrow +\infty$$.

Questions Assume we are given a sequence of polynomials $$(P_k)_{k\geq 2}$$ satisfying conditions 1,2,3 from above. Do we have $$\max_{x\in[0,1]} \mid P_k(x)\mid =O(\frac{1}{k^2})$$, for $$k$$ large enough?

I am sorry if this question is trivial, however analysis is not my field. This question is related to a question that I asked before.

Take $$P_k(x)=\frac{x^2}{k(k+1)}+\frac{x^2(1-x)}k.$$