One way of seeing it:
Absolute value is a continuous function so
Where it was used the fact that since when then as well, we have
In the following proof, why does the fact that imply that ?
Certainly i dont understand how that step is done,
in fact it would seem to me that doesnt make any sense since the left side does not depend on x.
Returning to the proof, earlier we had that
for all n if
Taking and using the continuity of the abs value we get that since if then ,
in our case :
if then .
And we end up having from where the proof follows quite easily.