The proper time (that is, the time the traveler measures on a clock) goes to zero very quickly towards the end. Even with both axes plotted logarithmically, it's just a sudden drop at the end towards light speed.

- [pic](https://home.davidgoffredo.com/hackernews/proper-time-one-li...)

- [plot](https://home.davidgoffredo.com/hackernews/proper-time-one-li...)