In my text, it says, "if cauchy sequence in a normed vector space converge, i.e. $$\lim_{j,k \to\infty} ||u_j - u_k|| = 0$$ then the normed vector space is complete".
The definition of completeness confuses me. Should I view $u_j$ and $u_k$ as completely different functions? Or should I view them as different instances of the same function $u$?
Just to understand this from an applied perspective, does it simply mean that a function reaches a steady state at some point?
$u_j$ and $u_k$ are elements of the normed vector space. If the vector space is for instance $C([0,1])$, the space of continuous functions on the interval $[0,1]$, twoexaples of sequences $\{u_k\}\subset X$ are $$ \{\cos(k\,x)\}_{k=1}^\infty,\quad\Bigl\{\frac{k}{k+x}\Bigr\}_{k=1}^\infty $$