[delusional]

This is obviously demonstrably true. A Turing running in O(n) time must halt. The one in O(n) space is free not to.

[thaumasiotes]

Almondsetat's proof seems more obvious. Given O(n) time, you can only use O(n) space, so you're comparing "O(n) space, any amount of time" with "O(n) space, O(n) time", and it turns out you get more resources the first way.