### 31st Putnam 1970

Problem B3

S is a closed subset of the real plane. Its projection onto the x-axis is bounded. Show that its projection onto the y-axis is closed.

Solution

Let Y be the projection onto the y-axis and X the projection on the x-axis. Let { yn} be any Cauchy sequence in Y. Then { yn } must converge to some real y. We have to show that y is in Y. For each n take any xn such that (xn, yn) is in S. Then { xn } lies in X, which is bounded, so { xn } is bounded. But any bounded sequence has a convergent subsequence. So we can take a subsequence { un } of { xn } which is convergent and therefore Cauchy. Let { vn } be the corresponding subsequence of { y n }. Then { vn } is also Cauchy and hence the sequence of points Pn = (un, vn) in S is Cauchy. But S is closed so P converges to some (u, v) in S. Hence { vn } converges to v, which is in Y (it is the projection of (u, v) ). But since { yn } converges to y, its subsequence { vn } must also converge to y. Hence y = v and y is in Y.