I had to prove for a homework assignment this function
$$ s_n(x) = \sum_{i=0}^n (-1)^i \frac{ x^{2i+1}}{(2i+1)!} $$
is a Cauchy sequence with respect to the sup norm for $$ s_n : [-M,M] \to \mathbb{R} $$ where M is any positive real number.
The goal is to show this sequence converges uniformly to the sine function, but the theorem I want to use requires $ s_n $ to be continuous, which I think is obvious because it is a power series and therefore a sum of real valued polynomials. But I wanted to know if there was a direct relationship between having a Cauchy sequence on a power series that defines a function and the function itself being continuous.
Thank you.