The following problem is taken from 'Real Mathematical Analysis' by Pugh, $2$nd edition, page $366$, exercise $3.$
Question: Let $T:V \rightarrow W$ be a linear transformation between normed spaces. Show that $$\| T \| = \sup\{ |Tv|: |v|<1 \} = \sup\{ |Tv|: |v| \leq 1| \} = \sup\{ |Tv|: |v| =1 \} = \inf\{ M:v \in V \Rightarrow |Tv| \leq M |v| \}.$$
Definition of $\| T \| = \sup \{ \frac{|Tv|}{|v|}: v \neq 0 \}.$
I managed to show that $\| T \| \leq \sup\{ |Tv|: |v \leq 1| \} \leq \sup\{ |Tv|: |v| =1 \} \leq \inf\{ M:v \in V \Rightarrow |Tv| \leq M |v| \} \leq \| T \|.$
However, I have no idea on how to show $\| T \| \leq \sup\{ |Tv|: |v|<1 \}.$
Any hint would be appreciated.