proving that a function that's continous on $[a, +\infty$) with an horizontal or oblique asymptote (at +$\infty$) is uniformally continious

Question

here's how I tried this (I guess it must be wrong), let's firstly prove the part about the horizontal one,

if there exists an horizontal asymptote $\implies \lim_{x\to +\infty}f(x)=k\in \mathbb{R} $ therefore, for the Heine-Cantor theorem the function f is uniformally continous on $[a,k] \forall x \in \mathbb{R}$ , i'm stuck in here on how to prove it in $(k,+\infty)$