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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.

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

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