Sequences

Concepts

• Convergent sequence 10 exercisesA sequence (a_n) converges to a number L if, for every tolerance ε>0, there is an index N from which all terms are within distanc…
• Uniqueness of limits 10 exercisesUniqueness of limits says that a sequence (a_n) cannot converge to two different numbers L and M; that is why it makes sense to s…
• Monotone sequence 10 exercisesA real sequence (a_n) is monotone when, as the index n grows, its terms either never decrease or never increase.
• Monotone convergence theorem 10 exercisesEvery real sequence (a_n), indexed by the natural numbers mathbb N, that is monotone and bounded converges to a real number.
• Subsequence 10 exercisesA subsequence of a sequence (a_n) is a sequence (a_n_k) made of terms of (a_n) with strictly increasing natural indices n_k.
• Bolzano-Weierstrass theorem 10 exercisesEvery bounded real sequence (a_n) has at least one subsequence (a_n_k) that converges to a number in the set mathbb R of reals.
• Cauchy sequence 10 exercisesA sequence (a_n) is Cauchy if, for every ε>0, there is an index N beyond which any two terms are less than ε apart.