Let $I = [ a , b ]$ where $a < b$, and let $f(x) = \exp(x)$ for all $x \in I$. Show that there exists a sequence of polynomials $\left( Q _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ that converges uniformly towards $f$ on $I$ and such that, for all $n \in \mathbb { N } ^ { * }$, the function $Q _ { n }$ does not coincide with $f$ at any point of $I$, except possibly at zero: $$\forall n \in \mathbb { N } ^ { * } , \quad \forall x \in I \backslash \{ 0 \} , \quad Q _ { n } ( x ) \neq \exp ( x ).$$
Let $I = [ a , b ]$ where $a < b$, and let $f(x) = \exp(x)$ for all $x \in I$. Show that there exists a sequence of polynomials $\left( Q _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ that converges uniformly towards $f$ on $I$ and such that, for all $n \in \mathbb { N } ^ { * }$, the function $Q _ { n }$ does not coincide with $f$ at any point of $I$, except possibly at zero:
$$\forall n \in \mathbb { N } ^ { * } , \quad \forall x \in I \backslash \{ 0 \} , \quad Q _ { n } ( x ) \neq \exp ( x ).$$