Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0000 362648 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0453 1212152 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0453 1212152 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1119 4105824 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1133 4133568 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1140 4151280 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1140 4168288 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1147 4176024 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1147 4176024 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1620 4415904 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1634 4433304 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1639 4454232 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.1722 4526792 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.1722 4527048 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

A Condition for Cauchy Sequences to Converge to the Same Limit

Theorem

Ifare Cauchy sequences in a metric spacesuch thatforthen

  1. is also a Cauchy sequence

  2. converges toif and only ifconverges to

Proof

For 1:

Applying the triangle inequality,

LetWe can findsuch that

Sinceis Cauchy, there existssuch that

Now letwe gethence is Cauchy.

For 2:

Using the triangle theorem again gives

Hence,but

Ifthen and

Similarly, ifthen