2014 Mac Book Air Stuck at 10.9.5

The Mavericks system has no further update within the 10.9.(5) range; the next

full system still available that would be path forward, is El Capitan OS X 10.11.6.

Instructions on how to get the downloadable installer (this is an installer app)

which then goes further online, to get actual system. Can be acquired through

steps (as described) ~ these details need to be followed, from support page:

• How to upgrade to OS X El Capitan (10.11.6) - Apple Support

https://support.apple.com/en-us/HT206886

Your MacBook/Air likely could require the El Capitan system to be installed

and running, before the Mac could see a later upgrade to newer system.

These would likely require El Capitan running before they'd later install:

• How to upgrade macOS High Sierra - Apple Support

https://support.apple.com/en-us/HT208969

• How to upgrade to macOS Mojave - Apple Support

https://support.apple.com/en-us/HT210190

At this time, options after 10.11.6 El Cap, are High Sierra 10.13.6; Mojave 10.14.6..

..Then perhaps Catalina 10.15.4+. (Because other issues and steps with Catalina

make that problematic, most Macs may require 'hardware upgrade' to run.)

You need to tell us exactly what does or doesn't happen.

Does your MBA see any updates?

If so what happens when you try to update?

If it doesn't see any updates, what happens when you try to manually check?