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Exercise 1.7: Ceiling Function and Sequences From Below

Constructing sequences that approach real numbers from below

Part 1: Existence of the Ceiling

For any real a, there exists m ∈ ℤ such that m < a ≤ m + 1.

012345m = 2m+1 = 3a = 2.70a ∈ (2, 3] means 2 < a ≤ 3

Key insight

The integers partition ℝ into intervals (m, m+1]. Every real number falls into exactly one such interval. The integer m is ⌈a⌉ - 1 (one less than the ceiling).