Proof.
This is true, either when the limit is finite or is infinite. We will
develop here the proof for the finite case, the reader can do the work
for the infinite case (
or
need the same
tools). Without loss of generality, we suppose that
.
Suppose that the function
tends to both real numbers
and
, when
is arbitrary close to
, i.e.
and
Denote
. We have:
Thus:
i.e.
By the triangular inequality, we have:
Hence, the non negative number
is less than any
positive number, so it must be equal to 0; this means that
.