1)This was a competition problem, though it was not posed exactly as I posted it. We will argue by induction, first for

this is clearly true. Now say it is true for

and we want to show it is true for

people. Out of all the distances amoing the soldier consider the minimal distance. This distance is determined by two soldiers, call them

and

. By the problem

must shoot toward

and

must shoot toward

. Now there are two possibilities. 1)Somebody shoots at them (those two soldiers). 2)Somebody does not shoot at them. If #1 then one of those soldiers is hit twice, so it is impossible for all the soldiers to be hit because the number of people exceeds the number of bullets. And if #2then we have

people shooting amongst themselves. Hence it is as if those two soldiers have nothing to do with them. And hence by induction someone remains alive.

2)I saw this amazing sequence problem on a forum. He is how I would do it:

Infinite series is the answer here.

Since

is convergent then

for some

. Choose

so large that

. Then for all

we have:

.

Now,

(by Uniform Convergence) since

.

Since

is continous by definition we see that this converges to

.