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Suppose that $(a_n)$ and $(b_n)$ are sequences of real numbers such that $a_n x + b_n \rightarrow ax+b$ for all $x\in\mathbb{R}$. Then my question is, is it necessarily true that $a_n\rightarrow a$? Or is it possible for $(a_n)$ to be a divergent sequence?

My question would be answered if a sequence of linear functions converges pointwise to a linear function if and only if it converges uniformly, but I’m not sure if that’s true. I’m guessing it is true.

I ask because this seems to be part of the intuition behind the idea that the slope of the tangent line is the limit of the slopes of the secant lines.

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Taking $x=0$ we have $b_{n}\rightarrow b$. Taking $x=1$ we have $a_{n}+b_{n}\rightarrow a+b$, then by basic limit rule we have $a_{n}\rightarrow a$.

For uniform convergence, it is not. For $f_{n}(x)=\dfrac{1}{n}x$, $f_{n}(x)\rightarrow 0$ but $\sup_{x}|f_{n}(x)|=\infty$.

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  • $\begingroup$ Does $a_n x + b_n \rightarrow f(x)$ for all $x\in\mathbb{R}$ imply that $f$ is linear? $\endgroup$
    – Keshav Srinivasan
    Commented Nov 14, 2019 at 1:18
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    $\begingroup$ So taking $x=0$ we have $b_{n}\rightarrow f(0)$. And taking $x=1$ we have $a_{n}+b_{n}\rightarrow f(1)$, by basic limit rule again we have $a_{n}\rightarrow f(1)-f(0)$, so with $x$ fixed, $f(x)=(f(1)-f(0))x+f(0)$. $\endgroup$
    – user284331
    Commented Nov 14, 2019 at 1:22
  • $\begingroup$ OK, thanks for your help. $\endgroup$
    – Keshav Srinivasan
    Commented Nov 14, 2019 at 1:34

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