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 3 of 5: d_1 is positive
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d1=(a0+a1)−1⋅a1=a0>0d_1 = (a_0+a_1) - 1\cdot a_1 = a_0 > 0
Detailed analysis

Directly, d1=(a0+a1)−a1=a0d_1=(a_0+a_1)-a_1=a_0, which is positive since a0a_0 is a positive integer.