I am trying to analyze the stability of finite difference schemes and encountered a theorem stating a scheme is stable iff
$$ \forall h>0\exists k_0,c_0>0, \forall 0<k\leq k_0, \forall l(l\in \mathbb{Z}\longrightarrow |G(2\pi l, k,h)|\leq1+c_0k)\tag{1} $$, where $|G(2\pi l, k,h)|$ is the amplication factor.
I can easily get a scheme is unstable iff $$ \exists h>0\forall k_0,c_0>0, \exists0<k\leq k_0, \exists l(l\in \mathbb{Z}\wedge |G(2\pi l, k,h)|>1+c_0k)\tag{2} $$
My question is is the first theorem equivalent to $$\forall h, k>0,l\in \mathbb{Z},|G(2\pi l, k,h)|\leq1 \tag{3}$$
$(3) \implies (1)$ is obvious, now I prove $(1) \implies (3)$, it is sufficient to check $(4) \implies (2)$, $(4) $is given below $$\exists h', k'>0,l'\in \mathbb{Z},|G(2\pi l', k',h')|>1 \tag{4}$$
let $h=h',\forall k_0,c_0$, let $k<\frac{|G(2\pi l', k',h')|-1}{c_0}$, then we have $\exists l=l'\wedge |G(2\pi l', k',h')|>1 \Box$