brownian motion in stopping time

Hello everyone!
I am stuck with a little problem concerning BM and stopping time.
If we define tau_n:=inf{t>=0 : |W_t|>= n}$, does anyone have idea how to show that |W(tau_n)|=n? I found one comment that on random interval [0,tau_n], Brownian motion is bounded by n, but it doesn't include any comment whatsoever.
Thank you very much!

Since BM has continuous sample paths, and variance equal to \(\displaystyle t\) (in your formulation), it would seem that \(\displaystyle |W(\tau_n)| = n\), almost by definition. There may be a tacit assumption that \(\displaystyle n\) is an integer, but that doesn't appeaqr necessary because of path continuity.

Continuity of sample paths is enough for this statement. Thank you very much for your help!