NDLI: A Note on Double Limits

Content Provider	Scilit
Author	Walsh, C. E.
Copyright Year	1927
Description	Let $u_{mn}$, $v_{mn}$ be functions of m and n, $v_{mn}$ being real and positive for all positive values of m and n. Suppose that either $v_{mn}$ increases steadily to infinity with n, or that $u_{mn}$ both tend to zero (the latter steadily) as n → ∞, for any fixed value of m. Denote by $w_{mn}$, and assume that $w_{mn}$ exists for every value of m, being denoted by $l^{m}$. Then from Stolz' extension of a result proved by Cauchy, and an allied theorem, we have , for all values of m. It follows from Pringsheim's Theorem that if the double limit of exists, being l, then $l_{m}$ → l as m → ∞.
Related Links	https://www.cambridge.org/core/services/aop-cambridge-core/content/view/629BAB854335F42CF63B8D2DD801B90D/S0013091500013584a.pdf/div-class-title-a-note-on-double-limits-div.pdf
Ending Page	198
Page Count	2
Starting Page	197
ISSN	00130915
e-ISSN	14643839
DOI	10.1017/s0013091500013584
Journal	Proceedings of the Edinburgh Mathematical Society
Issue Number	4
Volume Number	1
Language	English
Publisher	Cambridge University Press (CUP)
Publisher Date	1927-05-01
Access Restriction	Open
Subject Keyword	Proceedings of the Edinburgh Mathematical Society
Content Type	Text
Subject	Mathematics

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A Note on Double Limits