# Why is $\sup_{t \in A} |x_m(t)-x(t)| \leq \sup_{t \in A} \lim_{k \rightarrow \infty} |x_m(t)-x_k(t)|$?

Why is $$\sup_{t \in A} |x_m(t)-x(t)| \leq \sup_{t \in A} \lim_{k \rightarrow \infty} |x_m(t)-x_k(t)|$$?

$$(x_n)$$ is Cauchy, it's in $$BoundedLinear(A,\mathbb{K})$$, $$t \in A$$.

I'm thinking that this assumes that the sequence's distances near each other and thus one can assume that $$x_k(t)$$ will remain "more distanced" from $$x_m(t)$$ than the limit point $$x_k \rightarrow x$$ itself, since $$|x_m(t)-x(t)|$$ would be where the distance is smallest. But this sounds a bit inexact.

• What is $x(t)$? – angryavian May 9 at 18:15