Why must there be a sequence which converges to (m,b) if they are the arguments of a continuous function, but not a local maximum?

Question

I have been reading the paper "Monty Hall game: a host with limited budget" and trying to follow the arguments the authors make. However, I have been struggling to understand what they outline as Lemma 4.1.

In particular, they state:

If $g(m,b) = \beta \le 1$, then $(m,b) \ne (1,1)$. Since $g$ is continuous and $(m,b)$ is not a local maxima, there exists a sequence $ \{(m_n,b_n)\}_{n \ge 1}$ which converges to $(m,b)$ with $g(m_n,b_n) > \beta$. In particular, $f(m_n,b_n) > \alpha$.

The Lemma 4.1 itself is as follows:

For any $\alpha \in \left[ \frac{1}{3}, \frac{2}{3} \right]$, let:

$(m^\ast(\alpha),b^\ast(\alpha)) = \mathop{\arg\max}\limits_{f(m,b) \le \alpha} \ g(m,b)$

$(\tilde{m}(\beta),\tilde{b}(\beta)) = \mathop{\arg\min}\limits_{g(m,b) \ge \beta} \ f(m,b)$

Then $(m^\ast(\alpha),b^\ast(\alpha)) = (\tilde{m}(\beta),\tilde{b}(\beta))$.

They also define:

$g(m,b) = \frac{m}{3} + \frac{2b}{3}$

$\beta = g(m^\ast(\alpha),b^\ast(\alpha))$

and $f(m,b)$ is another, much longer function that would likely be better interpreted from the paper than if I try to type it out here.

I would like to understand several things:

• How do we know that $(m,b)$ is "not a local maxima" given that $g(m,b) = \beta \le 1$?
• Why does this, along with the continuity of $g$, imply the existence of a sequence as described?

I'm not a mathematician, more of an enthusiast. I feel very lost in this section of the paper and would really appreciate any pointers to improve my understanding of whatever it is I'm not grasping.

Answer 1

If $(m,b)$ is not a local maximizer of $g$ then for every ball around $(m,b)$, say of radius $1/n$, there exists a point $(m_n, b_n)$ for which $g(m_n, b_n)>g(m,b)$. This is what it means to not be a local max. Otherwise there would be some neighborhood where $g(m,b)$ dominated.