Proof
(1). Denote the limits of the sequences by and , respectively. Let
be given. Due to the convergence of the first sequence, there exists for
-
some such that for all
the estimate
-
holds. In the same way there exists due to the convergence of the second sequence for
some such that for all
the estimate
-
holds. Set
-
Then for all
the estimate
holds.
(2). Let
be given. The convergent sequence is
bounded,
due to
fact,
and therefore there exists a
such that
for all
.
Set
and .
We put
.
Because of the convergence, there are natural numbers
and
such that
-
These estimates hold also for all
.
For these numbers, the estimates
hold.
For the other parts, see
exercise,
exercise
and
exercise.