You can often show proof of onward travel on your phone, but don't rely on a live webpage as your only copy. Save the actual booking confirmation as a readable PDF or offline file, make sure the passenger and flight details are visible, and carry a printout if your airline, visa process, or border checkpoint asks for one. A reservation is useful only if you can open it when somebody needs to see it.
Picture this: you're at an airport check-in desk, the agent asks when you're leaving your destination, and your email app decides it needs to sign in again. Airport Wi-Fi drops. The message with your booking reference is buried under a dozen hotel promotions. None of this is a flight problem. It's a document-access problem, and it's avoidable.