A basic doubt on uniform convergence

Question

If $\{ln( P_n(x))\}$ converges uniformly to zero function does that imply that $\{P_n(x)\}$ uniformly converge to $1$ ? Why ? Here $\{P_n(x)\}$ is a sequence of polynomial function.

Answer 1

Yes, this is true, and the assumption that $P_n(x)$ be polynomial functions is irrelevant. To reframe the question, if $\{Q_n(x)\}$ is a sequence of functions which converge uniformly to zero, then $\{e^{Q_n(x)}\}$ converges uniformly to one.

This follows from the fact that $e^t$ is differentiable at $0$: Let $\delta>0$ be small enough so that for $|t-0|<\delta$, $\left\lvert\frac{e^t-1}{t} - 1\right\rvert < \frac{1}{2}$, so that $\frac{1}{2} < \frac{e^t-1}{t} < \frac{3}{2}$, which implies $\frac{1}{2} |t| < |e^t-1| < \frac{3}{2}|t|$. For arbitrary $\epsilon>0$, if $N$ is large enough so that for $n>N$, $|Q_n(x)|<\min\{\delta, \frac{2}{3}\epsilon\}$, then it follows that $|e^{Q_n(x)}-1|<\frac{3}{2} |Q_n(x)| < \epsilon$.