Let $\phi\_1,...,\phi\_n,...$ be a sequence of real-valued functions so that $\phi\_j:[0,1)\to[0,1)$, $\phi\_j(0)=0$, and $\phi\_j(\delta)$ converges to 0 as $\delta$ approaches 0 from the right for all $j\ge1$. Further suppose that $\sum\_{j=1}^\infty \phi\_j(\delta)$ **converges** and moreover, is strictly **smaller** than $1$ for an **arbitrary** $\delta$ in $(0,1)$.




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**MY QUESTION** is:


Does it hold that $\lim\_{\delta\to0^+} \sum\_{j=1}^\infty \phi\_j(\delta)=0$ ? If it doesn't, then does it turn out to be true if we in addition require that $\lim\_{\delta\to0^+}{\phi\_j(\delta) \over \delta}=0$ for all $j\ge1$?




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The sequence satisfying $\phi\_j(\delta)=\delta^j$ for all $j\ge1$, which gave me some kind of intuition, does support the equality, but I guess the equality does not hold true in general and I've been stuck in disproving it for days... Now I **really** need someone to show me a complete and detailed proof or disproof of the equality... Any help would be appreciated!



Convergence of the sum of a family of real-valued functions
Let $\phi\_i'(x)$ be defined as $1/2$ for $1/2^i<x<1/2^{i-1}$ and 0 elsewhere.


Clearly $\phi\_j':[0,1)\to[0,1)$, $\phi\_j'(0)=0$, and $\phi\_j'(\delta)$ converges to 0 as $\delta$ approaches 0 from the right for all $j\ge1$. Also $\sum\_{j=1}^\infty \phi\_j'(x)$ converges and is equal to 0 if $x=0$ and $1/2$ elsewhere.


This is therefore a counterexample to your assertion since $\lim\_{\delta\to0^+} \sum\_{j=1}^\infty \phi\_j(\delta)=1/2 \neq 0$.


It also shows that adding the further condition $\lim\_{\delta\to0^+}{\phi\_j(\delta) \over \delta}=0$ for all $j\ge1$ doesn't help since each $\phi\_i'$ is identically $0$ in $[0,1/2^i)$ which is a right neighbourhood of $0$.


An obvious condition that does work is to assert that the series $\sum\_{j=1}^\infty \phi\_j'(x)$
[converges uniformly](https://en.wikipedia.org/wiki/Uniform_convergence#To_series) at $0$. This is because $\phi\_i$ is continuous at $0$ and if $\sum\_{j=1}^\infty \phi\_j'(x)$ also converges uniformly at 0 then it must also be continuous at $0$ and hence $\lim\_{\delta\to0^+} \sum\_{j=1}^\infty \phi\_j(\delta)=\sum\_{j=1}^\infty \phi\_j(0)=0$.


