Unformatted text preview: Preliminary Exam
21 August 1999 . Use 6 = 2 +— §1T + 1°37 +    to prove that e is irrational. . b
. Let ambn 2 0, assume that San converges and that limsupn_,0° —n S M < 00 show that n
2 bn converges. . Let f be bounded on the real interval ((1,1)), show that if in addition f is both continuous and
monotone then f is uniformly continuous. . Prove that f is integrable on . Define f(:r) = {
[0,1]. 0 ,x irrational
% ,x = m/n where m and n relatively prime . Let {fn} be a sequence of uniformly bounded Riemann integrable functions on [0,1], set Fn(s) 2 / fn(t)dt for 0 S s g 1. Prove that a subsequence of {Fn} converges uniformly on
0 [0,1]. . Let f (x) be a differentiable mapping of the connected open subset V of IR". Suppose that
f’(m) = 0 on V, prove that f is constant on V. . Let f(:r, y) = (u, v) where u = x2 —y2 and v = 2333; describe a map from R2 to R2. (a) What is
the range of this map? (b)Show that if (u, v) 75 (0,0) then f has an inverse in a neighborhood
of (11,11). (c) Show that there is no neighborhood of (0,0) in which f has an inverse. ...
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 Spring '11
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