# How to show that if $\inf A > - \infty$ and $\inf B > - \infty, A, B \subset \mathbb{R}$ then $A+B = \{a+b:a \in A, b \in B \}$ is bounded below?

How to show that if $\inf A > - \infty$ and $\inf B > - \infty$, $A,\, B \subset \mathbb{R}$ then $A+B = \left\{a+b:a \in , b \in B \right\}$ is bounded below?

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... bounded from below? – Hagen von Eitzen Jan 16 '13 at 16:21
What would be the most obvious bound from below you might think of? – Thomas Andrews Jan 16 '13 at 16:33
If $inf A = a$ and $inf B = b$, then $a \leq x, \forall x \in A$ and $b \leq x, \forall x \in B$ so bound from below for $C=A+B$ is a+b, which satisfies $a+b \leq x, \forall x \in C$ – alvoutila Jan 16 '13 at 16:43
How is this measure theory? – mrf Jan 16 '13 at 16:58
They use $inf A$ for measure $m_n$ – alvoutila Jan 16 '13 at 17:00

Denote $m_A=\inf{A},\;\; m_B=\inf{B}.$ Then \begin{gather} \forall{a}\in{A} & {a}\geqslant{m_A} \\ \forall{b}\in{B} & {b}\geqslant{m_B} \end{gather} So, \begin{gather} (\forall{a}\in{A})(\forall{b}\in{B}) \\ {a+b}\geqslant{m_A+m_B} \end{gather}
Clearly $\inf A+\inf B\leq A+B$. Since the left is a lower bound, we have $\inf A+\inf B\leq\inf (A+B)$.