Claim: For any $A,B\subseteq\mathbb{R}$, $\inf(A+B)\leq \inf(A) + b$ for all $b\in B$.
Proof: Suppose for a contradiction, there exists $b\in B$ with $\inf(A+B)-b>\inf(A)$. Then $\forall a\in A$, $a=(a+b)-b\geq \inf(A+B)-b > \inf(A)$. Take $A=[0,1]$, then $0\in A$ and $0=\inf(A)$, but $0\ngtr 0$, thus contradiction.
But I feel like this proof is wrong?
The proof is not conclusive because you have realized to a particular example.
Rather, the proof can go by this way: For any $b\in B$ and $a\in A$, we have $a+b\geq\inf(A+B)$ and hence $a\geq\inf(A+B)-b$, since this expression is valid for all $a\in A$, we have $\inf A\geq\inf(A+B)-b$, move $b$ to the left-sided, we are done.