V=a Banach space over R
W=a proper closed subspace of V
Prove : For any $\epsilon > 0$, there is a v $\in$_V_ such that ||v||=1 and ||v+w||$\geq$ $1$ - $\epsilon$
I have shown that there exists such v $\in$ V such that ||v||=$1$
But I am having a trouble to show that ||v||=1 and ||v+w||$\geq$ $1$ - $\epsilon$
I was trying to use the fact that such v with ||v||=$1$ are the vectors on the unit sphere. But I do not know how to proceed with this idea.
So any ideas and opinions would be appreciated.