In the paper (or refer to this answer to view the full proof of this result):
Tsing N.K. [1984]. Infinite dimensional Banach spaces must have uncountable basis—an elementary proof. Amer. Math. Monthly, 96 (5), 505-506.
I cannot show in the very last line that $\displaystyle\frac{1}{2}t_m r_m$ does not approach to $0$ as $m$ increases, so we cannot guarantee that $\| u_{n} - u \|$ does not approach $0$. In fact $r_{m+1} \le r_m$ since we're taking the infimum of a set that is becoming larger. What am I missing here?