(a) Label the right edge of the circle point M, and the left edge point N.If the temperature at M is the same as that at N, then we have antipodal points where the temperature are the same.

But what if the temperature at M is lower than that at N?

Imagine a stick longer than the diameter of the circle, and its midpoint is fixed at the center of the circle. So the two contact points of the stick and the circle are antipodal points.Label the two points A and B. Now imagine the stick rotates from the x-axis anticlockwise, with A going  from M to N, and B going from N to M.It stop after rotating 180 degrees.So at the initial time, temperature at A is lower than temperature at B;temperature at B is higher than temperature at A.After the rotation, A has higher temperature than B.During the whole process the temperature can only change continuously, and the stick cannot be bent. Thus there must exist a certain instant, at which the temperature at A and that at B are the same.

(b) My argument rely implicitly on the continuity of the change of temperature. In real life, temperature changes continuously, and I use a real life way to interpret the proof, in which the stick is assumed to rotate smoothly (also related to continuity) .

1. The change of temperature as the altitude increases can be considered as a function. Generally, for every 100 meters’ increment at the altitude, the temperature decrease by 0.6℃. If we assume the temperature at the sea level is 25℃,the temperature(T,℃) vs altitude(x,m) function can be written as: y= -0.006x+25

Since it is a linear function, it does not have a horizontal asymptote. Assume it has a horizontal asymptote y=a

so the horizontal asymptote does not exit.

2. There is a pancake, its area is 1.If we cut it into halves.Then after the first time we cut it , the area of each of the pieces is 1/2.After the third time,the area of each of the pieces is 1/4.

…..

If we write down the area after every time we cut it ,it will be

1/2,1/4,1/8,……,

It does not converge.

3.If we sum up all the terms above,we get 1/2+1/4+1/8+…

By the geometric test,we know that the series converges to 1.