The well-known summability methods of Euler and Borel are studied as mappings from ℓ 1 into ℓ 1 . In this ℓ − ℓ setting, the following Tauberian results are proved: if x is a sequence that is mapped into ℓ 1 by the Euler-Knopp method E r with 0$"> r > 0 (or the Borel matrix method) and x satisfies ∑ n = 0 ∞ | x n − x n + 1 | n < ∞ , then x itself is in ℓ 1 .