Let  M_n×n be the space of real n×n matrices, regarded as a metric space with the distance function

\(\displaystyle d(A,B)=\sum_{i,j} |a_{ij}-b_{ij}|\)

for A=(a_ij) and B=(b_ij).
Prove that \(\{A\in M_{n\times n}: A^m=0 \text{ for some positive integer }m\}\) is a closed set.

The best solution was submitted by Gee Won Suh (서기원), 2009학번. Congratulations!

Here is his Solution of Problem 2010-10.

Alternative solutions were submitted by 정성구 (수리과학과 2007학번, +3), 김치헌 (수리과학과 2006학번, +3), 강동엽 (2009학번, +2).

Let M>0 be a real number. Prove that there exists N so that if n>N, then all the roots of \(f_n(z)=1+\frac{1}{z}+\frac1{{2!}z^2}+\cdots+\frac{1}{n!z^n}\) are in the disk |z|<M on the complex plane.

The best solution was submitted by Jeong, Seong Gu (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2010-9.

Alternative solutions were submitted by 최홍석(화학과 2006학번, +3), 김호진(2009학번, +3), 김치헌 (수리과학과 2006학번, +3).