We are given the following: $$w_s - w_u(2+r) - (1+r)h \geq 0 \tag{1}$$ where $h$ is positive.

Now, consider the following fraction: $$f = \frac{w_u(2+r) + (1+i)h - w_s}{i - r}$$

The condition for the entire fraction to be positive (assuming $i > r$) is:

$$w_u(2+r) + (1+i)h - w_s \geq 0 \tag{2}$$

Combining (1) and (2) leads to the following:

$$(i-r)h \geq 0$$ which leads me to think that $f = 0$ when $(i-r) = 0$.

However, I am having trouble proving this formally, since if $i=r$, the fraction is undefined. I am unsure how to take the limit of this (applying L'Hopital's rule did not yield the result).

How do I show formally that $f \to 0$ as $i \to r$?


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