Let $\| \cdot \|$ be matrix norm induced by a vector norm $\| \cdot \| \text{.}$ Prove that for any $A \in \Cmxm \text{,}$ the spectral radius, $\rho( A )$ satisfies $\rho( A ) \leq \| A \|\text{.}$
Some results in linear algebra depend on there existing a consistent matrix norm $\| \cdot \|$ such that $\| A \| \lt 1 \text{.}$ The following exercise implies that one can alternatively show that the spectral radius is bounded by one: $\rho( A ) \lt 1\text{.}$
Given a matrix $A \in \Cmxm$ and $\epsilon \lt 0 \text{,}$ there exists a consistent matrix norm $\| \cdot \|$ such that $\| A \| \leq \rho( A ) + \epsilon \text{.}$