If $a+\epsilon > b$ for each $\epsilon >0$, can we conclude that $a>b$?
Please help me to clarify the above. Thanks in advance.
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No, you cannot: for every $\epsilon>0$ it’s true that $1+\epsilon>1$, but it’s not true that $1>1$. What you can conclude is that $a\ge b$. To see this, suppose that $a<b$, and let $\epsilon=b-a$; then $\epsilon>0$, but $a+\epsilon=a+(b-a)=b\not>b$, contradicting the hypothesis. Thus, it must be the case that $a\ge b$.