This does not seem too difficult, but I've been stuck here for a while. Could someone give me a hint?
Question: Let $A$ be a linear transformation on $R^n$. $v$ is a vector in $R^n$. Prove that if $\frac{||Av||}{||v||}\rightarrow0$ as $v\rightarrow 0$, then $A=0$. $||\cdot||$ denotes Euclidean norm.
My try: proof by contradiction. Suppose at least one element in $A$, namely $A_{pq}$, is nonzero. Ideally, the contradiction would be the limit of the above fraction is nonzero. The problem is the numerator always approaches zero as $v\rightarrow 0$. Is there any other approach?
For any $v$ with $||v||\ne0$, $$ ||Av||=||v||\,\frac{||Av||}{||v||}=||v||\,\frac{||\lambda Av||}{||\lambda v||}=||v||\,\frac{||A(\lambda v)||}{||\lambda v||} \to 0 $$ as $\lambda\to0$. Hence $Av=0$.