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We are asked to show that

  1. $B$ is the domain of $Q$ and is compact
  2. $Q(\beta)$ is continuous in $\beta$
  3. $Q(\beta)$ is uniquely minimized at $\beta_0$

implies

$\forall \epsilon >0 \; \exists \delta>0 : \underset{||\beta - \beta_0||>\epsilon}{\inf} Q(\beta) \geq Q(\beta_0) + \delta$.

Why do I need conditions 1. and 2. ? Am I not immediately finished using condition 3.?

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I think condition $3$ is just meant to say that if the minimizer $\beta_0$ exists, then it is unique.

The first two conditions can be used to show that the minimizer exists.

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  • $\begingroup$ yes, that was it. $\endgroup$ – stollenm Dec 19 '17 at 14:13

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