Implication of an inequality relation Suppose I have linear function $f$ on $\mathbb{R}^n$ and another function $p$, which is positive homogeneous, i.e. $p(\lambda x)=\lambda p(x)$ for all $\lambda\ge 0$. We have the following implication
$$f(x)>-1\Rightarrow p(x)>-1$$
Since we can multiply by positive scalars, we get
$$f(x)>-y\Rightarrow p(x)>-y$$
for every $y\ge 0$. I thought this should imply that $p(x)\ge f(x)$, but my lecture note says the opposite: $p(x)\le f(x)$. The problem is, with the latter the whole rest of the proof does not work anymore! So where is my error in reasoning? 

"Since we can multiply by positive scalars": what does it mean?
Is this something like: for $x = λy$ with $λ>0$ and y in $ℝ^n$:
$f(x)>−1 ⇒ p(x)>−1$
$f(λy)>−1 ⇒ p(λy)>−1$
$λf(y)>−1 ⇒ λp(y)>−1$
$f(y)>−1/λ ⇒ p(y)>−1/λ$
?
