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Suppose that $(c_{j, n})$ is a real infinite dimensional triangular array where $1 \leq j \leq n$ with the properties that $\max\limits_{1 \leq j \leq n } |c_{j, n}| \rightarrow 0$ and $\sum\limits_{j=1}^n c_{j, n} \rightarrow \lambda$ when $n\rightarrow\infty$, and $\sup\limits_{n\in\mathbb{N}} \sum_{j=1}^n |c_{j, n}|<\infty.$

Please help me to prove that therefore $\prod\limits_{j=1}^n(1+c_{j, n})\rightarrow e^\lambda$.

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up vote 1 down vote accepted

Note that $$\tag1\ln\prod_{j=1}^n(1+c_{j,n})=\sum_{j=1}^n\ln(1+c_{j,n}).$$ As $\ln(1+x)=x-\frac12 x^2+\frac13x^3\pm\ldots$, we have $|\ln(1+x)-x|\le x^2$ if $|x|$ is small enough. For $x=c_{j,n}$ this is guaranteed by $\max_{1\le j\le n}|c_{j,n}|\to 0$. Therefore, the sum on the right side of $(1)$ differs (for $n$ large enough) from $\sum c_{j,n}$ by at most $$\left\vert \sum_{j=1}^n c_{j,n}^2\right \vert\le \max_{1\le j\le n} |c_{j,n}|\cdot \sum_{j=1}^n |c_{j,n}|\to 0.$$ Hence, $\ln\prod_{j=1}^n(1+c_{j,n})\to \lambda$ as was to be shown.

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