$\"\"$ \ \

The sequence is $\"\"$ . \ \

If the sequence $\"\"$ is bounded, monotonic and increasing then $\"\"$ . \ \

If the sequence $\"\"$ is bounded, monotonic and non increasing then $\"\"$ . \ \

Then the sequence $\"\"$ is bounded, monotonic and non decreasing sequence which converges. \ \

Therefore, $\"\"$ is converges hence the limit value of $\"\"$ exist. \ \

$\"\"$ \ \

$\"\"$ is converges hence the limit value of $\"\"$ exist.