Let
{
𝑎
𝑛
}
𝑛
=
0
∞
be a sequence. Let
𝐿
∈
𝑅
.

Which statements are equivalent to "
{
𝑎
𝑛
}
𝑛
=
0
∞
→
𝐿
" ?

Select all the correct answers.

Question

Let 
{
𝑎
𝑛
}
𝑛
=
0
∞
 be a sequence. Let 
𝐿
∈
𝑅
  .

Which statements are equivalent to " 
{
𝑎
𝑛
}
𝑛
=
0
∞
→
𝐿
 " ?

Select all the correct answers.

BlackTom AI · Accepted Answer

The question asks which statements are equivalent to the definition that the sequence a_n converges to L as n → ∞. In plain terms, this means: for every positive tolerance ε, there exists a point n0 after which all terms a_n stay within ε of L. Option 1: ∀ ε > 0, ∃ n0 ∈ N, ∀ n ∈ N, n ≥ n0 ⇒ |a_n − L| < ε. - This is exactly the standard formal definition of convergence. It says: no matter how small a positive ε you choose, you can find an index n0 after which all subsequent terms are within ε of L. This option correctly captures the convergence criterion. - Why this is correct: It matches the typical ε-N (epsilon-N) formulation of convergence, with the strict inequality < ε ensuring the terms get arbitrarily close to L but not necessarily equal to L at any finite stage. Option 2: ∀ ε ≥ 0, ∃ n0 ∈ N, ∀ n ∈ N, n ≥ n0 ⇒ |a_n − L| ≤ ε. - This changes ε to allow ε = 0 and uses ≤ instead of <. When ε = 0, the statement would require |a_n − L| ≤ 0 for all sufficiently large n, i.e., a_n = L eventually. Not all convergent sequences satisfy eventual equality; many converge to L without ever equaling it exactly. Therefore this version is not equivalent to convergence in general. This option is incorrect. - The use of ≤ also weakens the inequality in a way that, for ε > 0, is equivalent to < ε, but the inclusion of ε = 0 makes the statement stronger than the standard definition, hence not equivalent. Option 3: ∀ ε > 0, ∃ n0 ∈ N, ∀ n ∈ N, n ≥ n0 ⇒ |a_n − L| < ε. - This appears to be the same as Option 1 in substance: it states precisely that for every positive ε there exists an index after which all terms stay within ε of L. It is another correct form of the convergence criterion, just stated with a slightly different phrasing but without altering meaning. This option is correct. Option 4: ∀ ε ≥ 0, ∃ n0 ∈ N, ∀ n ∈ N, n ≥ n0 ⇒ |a_n − L| ≤ ε. - Like Option 2, this version allows ε to be 0 and uses ≤. The requirement becomes that eventually |a_n − L| ≤ 0, i.e., a_n = L for all large n, which is not guaranteed for a general convergent sequence. Hence this is not equivalent to the general convergence definition. This option is incorrect. In summary: - Options 1 and 3 accurately reflect the standard ε-N definition of convergence to L. - Options 2 and 4 fail to be equivalent because they permit ε = 0 and use ≤, imposing a stronger condition (eventual equality) that is not required by convergence in general.

MAT137Y1 LEC 20249: Calculus with Proofs (all lecture sections) Pre-Class Quiz 50 (11.1 and 11.2)

Let { 𝑎 𝑛 } 𝑛 = 0 ∞ be a sequence. Let 𝐿 ∈ 𝑅 . Which statements are equivalent to " { 𝑎 𝑛 } 𝑛 = 0 ∞ → 𝐿 " ? Select all the correct answers.

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