Convergence of Sequences, Rigorously
Prove convergence from the definition: for every tolerance there is a point in the sequence past which every term stays inside it. Limit laws become theorems with proofs rather than rules to apply.
What a learner can do afterwards
- Prove a stated limit by producing N for an arbitrary epsilon
- Prove that a limit is unique
- Prove one of the limit laws from the definition
1 · Read
Like 1 minus 1 over n climbing toward its ceiling 1, a sequence is an ordered list of numbers a1, a2, and so on. It converges to L when its terms get and stay arbitrarily close to L. Precisely, for each positive epsilon there is a cutoff N so that all terms past N sit within epsilon of L. Smaller epsilon may force a larger N.
To prove 1 over n tends to 0, start with an arbitrary epsilon and work backward. You need 1 over n below epsilon, which holds past N equal to 1 over epsilon. For epsilon 0.02, N equal to 50 does the job. Present that N and verify it: every later term is smaller than 0.02.
A convergent sequence has exactly one limit: if L and M both worked, their difference would have to be zero. The limit laws then split hard limits into easy pieces: the limit of a sum is the sum of the limits, and likewise for products and quotients. So if a_n tends to 2 and b_n tends to 5, their sum tends to 7. Oscillators like (-1) to the n never settle and diverge.
Name the limit with epsilon and N, prove it by working backward, and split the rest with the limit laws.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.