# University of Illinois at Urbana Champaign Fall

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University of Illinois at Urbana-Champaign Fall 2006 Math 444 Group E13 1. Prove by induction that for all n ≥ 1 one has 1 12 + 1 22 + . . . + 1 n2 < 2 . Correction. First, following the hint, let us call P (n) the property 1 12 + 1 22 + . . . + 1 n2 ≤ 2? 1 n . Then P (1) is the assertion 1 ≤ 1, so P (1) is true. Now, assume that n ? N is such that P (n) is true. We wish to deduce from this that P (n+1) is also true. We have 1 12 + 1 22 + . . . + 1 (n + 1)2 = 1 12 + 1 22 + . . . + 1 n2 + 1 (n + 1)2 ≤ 2? 1 n + 1 (n + 1)2 (?) by the induction hypothesis. To obtain what we want, it is therefore enough to prove that? 1 n + 1 (n + 1)2 ≤ ? 1 n + 1 , which is equivalent to 1 (n + 1)2 ≤ 1 n ? 1 n + 1 .

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##### Mathematical induction

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2 2 21 2 n n
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n∈N P(n) P(n+1)
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2 2 2 2 2 2 2 21 2 (n+1) 1 2 n (n+1) n (n+1)
1 1 1
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P(n+1) P(n) P(1)
P(n) n∈N
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n
n
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P(n) n≥ 2
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x∈R f g◦f f(x) = 0 x∈R g(x) =x
x∈R g g◦f
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2• x∈R h(x) = (x+2) h(x)≥ 0 x h(R)
√2[0,+∞) y≥ 0 x y = (x+2) x = y−2
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2x∈E 0≤x≤ 1 2≤x≤ 3 4≤h(x) = (x+2) ≤ 9 h(E)⊂ [4,9]

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