Let $Y$ and $Z$ be two closed subspaces of a Banach space $X$ with $Y\cap Z=\{0\}$.

I know that $Y+Z$ is a closed subspace of $X$ $\iff \exists α>0:\quad ∥y∥≤α∥y+z∥\forall y∈Y,\forall z∈Z$.

However, reading this question A criterion for the sum of two closed sets to be closed ? commenter posted that: the standard equivalence to the sum being closed is that the unit spheres of $Y$ and $Z$ are a positive distance apart i.e. $∃r>0\quad ∥y−z∥≥r\quad ∀y∈Y\,∀z∈Z\quad s.t.\quad ∥y∥=∥z∥=1$.

Could anybody provide me with a proof or rather a reference to where I can see the proof of this equivalence?

Let $Y$ and $Z$ be two closed subspaces of a Banach space $X$ with $Y\cap Z=\{0\}$.

I know that $Y+Z$ is a closed subspace of $X$ $\iff \exists \alpha > 0:\quad \lVert y\rVert \le \alpha\lVert y+z\rVert \forall y∈Y,\forall z∈Z$.

However, reading this question A criterion for the sum of two closed sets to be closed ?, Bill Johnson posted that: the standard equivalence to the sum being closed is that the unit spheres of $Y$ and $Z$ are a positive distance apart i.e. $\exists r>0\quad \lVert y−z\rVert ≥r\quad \forall y\in Y\,\forall z\in Z\quad s.t.\quad \lVert y\rVert=\lVert z\rVert=1$.

Could anybody provide me with a proof or rather a reference to where I can see the proof of this equivalence?