Suppose I have a normed space $(X, ||\cdot||)$ and a sequence of functions $\{f_n(x)\}_{n=0}^{\infty}$ on an interval $[a, b]$. I should mention that here $X$ is the space of functions (polynomials) on the interval $[a, b]$. How would I go about showing that $\{f_n(x)\}_{n=0}^{\infty}$ is a Cauchy sequence?
The definition of Cauchy in this context is: given an arbitrarily small value $\epsilon$, there exists a natural number $N$ such that for all $m, n \geq N$, $||f_m(x) - f_n(x)|| \leq \epsilon$
(In my specific problem, I know that my sequence converges to a known function).
My issue is that I'm not sure what I am able to use here. If I was able to use the fact that my sequence converges (generally speaking) to a function, I think I could do this exercise fairly quickly, but I'm not sure.
What advice can you give me here?
PS, I do not want to write too many details since this is a HW problem.
If $X$ is a metric space (as any normed space is), then any convergent sequence of contains elements which eventually all become arbitrarily close to some other point $X$.
Then you should be able to show that this means that the elements in the sequence eventually become close to each other on the basis of elementary properties of metric spaces.
(I have been slightly vague as per your request)