I'm doing the following problem, which is about the condition under which a function equals its Taylor series.
Taylor's Inequality. If $\left|f^{(N+1)}(x)\right| \leq B$ for all $x$ in the interval $[c-d, c+d]$, then the remainder $R_N(x)$ (for the Taylor polynomial to $f(x)$ at $x=c$ ) satisfies the inequality $$ \left|R_N(x)\right| \leq \frac{B}{(N+1) !}|x-c|^{N+1} \quad \text { for all } x \text { in }[c-d, c+d] \text {. } $$ If the right-hand side of Taylor's inequality goes to 0 as $N \rightarrow \infty$, then the remainder must go to 0 as well, and hence for those $x$ values, the function matches its Taylor series.
Question: Use Taylor's inequality to show that $e^x$ converges to its Taylor series at $0$ for all real $x$.
I've managed to show what was asked, and checked my answer against the solution. However, I don't understand why we only care about the Taylor series at $0$. I was thinking that we need to show convergence holds for all $c$.