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Showing 6 pages in Real Analysis.
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Proofs
Structured arguments in Proof Builder, with theorem statements, line-by-line reasoning, and nearby variations.
Worked Examples
Standard problems where the computation and the explanation stay tied to the live tool.
Explanations
Concept pages that build intuition first and then connect it to formal notation and exact calculations.
Proofs
A catalog of interesting and common proofs to learn, study, and revisit.
Proof that the limit of a convergent sequence is unique
A convergent sequence cannot settle down to two different numbers. The proof is short, but it is a good model for how contradiction and the definition of limit work together.
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Epsilon-delta proof that
This proof is common because it shows the standard move in early analysis: factor the expression, then bound the extra factor by forcing into a smaller interval first.
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Proof that every convergent sequence is bounded
This theorem is a standard follow-up to the definition of convergence because it shows how one tail estimate plus finitely many early terms gives a global bound on the whole sequence.
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Explanations
Concept pages that build intuition first and then connect it to formal notation and MCPCalc tools.
What an epsilon-delta proof is actually controlling
An epsilon-delta proof is a control problem: keep close enough to a point so the function value stays inside a target band around the limit.
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What convergence means for sequences
A sequence converges when its terms eventually stay as close as you want to a single number. The word eventually matters more than the first few terms.
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What convergence means for infinite series
An infinite series converges when its partial sums settle toward a finite number. The important object is the running total, which is why the convergent geometric series and the divergent harmonic series tell different stories even though both have terms that go to zero.
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