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
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 every positive epsilon there is an integer N so that every n greater than or equal to N has distance between a_n and L less than epsilon. Epsilon is chosen first, and N may depend on it. A geometric sequence c times r to the n with absolute r below 1 has powers that shrink to 0, so the same definition proves any such geometric sequence tends to 0.
To prove 1 over n tends to 0, start with arbitrary epsilon and work backward. Need 1 over n below epsilon, so choose N greater than 1 over epsilon, such as floor of 1 over epsilon plus 1. For epsilon 0.02, N equal to 51 works: every n at least 51 gives 1 over n at most 1 over 51, below 0.02. State N and verify the inequality.
A convergent sequence has one limit. If L and M both worked, choose a tail within epsilon over 3 of each; then distance between L and M is less than 2 epsilon over 3 for every epsilon, forcing L equal to M. For sums, choose tails with each error below epsilon over 2, then triangle inequality gives sum error below epsilon. Products and quotients need bounded factors and a denominator bounded away from zero. An oscillator such as (-1) to the n fails because no tail stays near one number. Constant multiples pull a fixed factor through the same epsilon bound.
For a rational expression such as (5n minus 2) over (n plus 7), divide numerator and denominator by n. It becomes (5 minus 2 over n) over (1 plus 7 over n). The small fractions tend to 0 by the sequence limit laws, leaving 5. This leading-power method works when numerator and denominator have the same degree.
Name the limit with epsilon and N, prove it by working backward, and split the rest with the limit laws.
Reading text was adapted in original wording from the listed records. Corpus license labels are not primary publisher evidence and are recorded as null. Extra items were rewritten with original provenance.
2 · Watch
Take it off screen
Where it sits
Learn first
Optional connections
This opens up
14 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.