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Let $1<b\in \mathbb{R}$ and $y\in \mathbb{R}$. I have proved that $A=\{x\in \mathbb{R}\mid b^x < y\}$ is nonempty when $y > 1$. Please give me any hint how to show that $A$ is nonempty when $y\leq 1$.

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It depends on your definition of $b^x$ – Norbert Jul 9 '12 at 20:44
I want to delete this post.. – Katlus Jul 9 '12 at 20:47
$x<0$ basically lets you convert $b$ to $1/b$ and use your previous proof. – gt6989b Jul 9 '12 at 20:50
If $y = 1$, there won't be $x \in \mathbb{R}$ s.t. $b^{x}<1$ for $b=1$. – data Jul 9 '12 at 20:53
@Katlus: Is there not a "delete" link in the bottom left of your post (by "link," "edit," etc.)? – Jonas Meyer Jul 10 '12 at 14:10
up vote 0 down vote accepted

Consider the sequences $x_n=-n$ and $b_n = b^{x_n}$. Ask what the limit of $b_n$ is and use facts about limits.

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