How do I stop previews of recent activity appearing when I double tap the home button

HHow do I stop previews of recent activity from appearing when I double tap the home button

As others have mentioned, you are seeing the Multitasking bar which is a feature of iOS:
Understand multitasking and background activity on your iPhone, iPad, or iPod touch - Apple Support
If left alone long enough and after running many apps over a period of time, iOS will eventually close off the app(s) that have not been used for the longest period of time to gain back memory to be used to run other apps that are currently not in Multitasking. There's no way to automatically prevent an app from appearing in the Multitasking bar nor any way to shut them all down at once. You can easily close them off manually as needed by swiping up on the app's displayed screen shot (NOT the lower icon). Multitasking is by design and cannot be changed/modified.

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