So I want to prove that $x'=f(t,x)$ has global solution on $\Bbb R$ if the follow $2$ conditions are satisfied:
$|f(t,x)|\le C(1+|x|)$ (linearly bounded) for some $C >0$ and
if $f(t,x)$ is locally Lipschitz.
I feel like I have to prove that $f(t,x)$ is Lipschitz so that the solution doesn't blow up and then it’s defined on $\Bbb R$. But how can I prove it rigorously? Does the Grönwall lemma help here?
From the Grönwall lemma you get that whenever $x$ is defined, $$ |x(t)|\le e^{C|t-t_0|}(1+|x(t_0)|)-1. $$ This bound on the values of the solution prevents any divergence in finite time to happen.
You can use this to extend any solution indefinitely by small patches, or you can use a global version of Picard-Lindelöf that uses a modified norm $$ \|x\|_L=\sup_{|t-t_0|\le T} e^{-2L|t-t_0|}|x(t)| $$ where $L\ge C$ is a Lipschitz constant on $[t_0-T, t_0+T]\times B(0,R)$ with $R=e^{CT}(1+|x_0|)$. The Picard iteration is contractive in this norm, allowing to apply the Banach fixed-point theorem on $C([t_0-T, t_0+T]\to B(0,R))$ directly.