MathLabs

Problem 1

Let a0<a1<a2<⋯a_0<a_1<a_2<\cdots be an infinite sequence of positive integers. Prove that there exists a unique integer n≥1n\ge1 such that an<a0+a1+⋯+ann≤an+1.a_n<\frac{a_0+a_1+\cdots+a_n}{n}\le a_{n+1}.
Step 4 of 5: The sequence d_n is strictly decreasing
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dn+1−dn=n(an−an+1)<0d_{n+1} - d_n = n(a_n - a_{n+1}) < 0
Detailed analysis

Compute dn+1−dn=[(a0+⋯+an+1)−(n+1)an+1]−[(a0+⋯+an)−nan]=n(an−an+1)d_{n+1}-d_n=\big[(a_0+\cdots+a_{n+1})-(n+1)a_{n+1}\big]-\big[(a_0+\cdots+a_n)-na_n\big]=n(a_n-a_{n+1}), which is negative since n≥1n\ge1 and an<an+1a_n<a_{n+1}. So d1>d2>d3>⋯d_1>d_2>d_3>\cdots is a strictly decreasing sequence of integers.