Guest2
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How does one prove that $$\displaystyle \lim_{x \to a} x^n = a^n$$ directly from the epsilon-delta definition?
$\displaystyle |x^n-a^n| < \varepsilon \implies \bigg|x-a\bigg|\bigg|\sum_{k=0}^{n-1}x^k a^{n-k-1}\bigg| <\varepsilon \implies |x-a| < \frac{\epsilon}{ \bigg| \displaystyle \sum_{k=0}^{n-1}x^k a^{n-k-1}\bigg|}$.Ackbach said:What have you tried so far?
Guest said:How does one prove that $$\displaystyle \lim_{x \to a} x^n = a^n$$ directly from the epsilon-delta definition?
Euge said:Looking back at Guest's initial work, I assume $n$ is a positive integer. Let $a$ be a fixed real number and suppose $0 < |x - a| < 1$. By the triangle inequality, $|x| = |(x - a) + a| \le |x - a| + |a| < 1 + |a|$. Again, by the triangle inequality,
$$\left|\sum_{k = 0}^{n-1} x^ka^{n-k-1}\right| \le \sum_{k = 0}^{n-1} |x^k a^{n-k-1}| = \sum_{k = 0}^{n-1} |x|^k|a|^{n-k-1} < \sum_{k = 0}^{n-1} (1 + |a|)^k (1 + |a|)^{n-k-1} = n(1 + |a|)^{n-1},$$
and thus
$$|x^n - a^n| < n(1 + |a|)^{n-1}|x - a|,$$
which can be made less than a positive number $\epsilon$ by making
$$|x - a| < \frac{\epsilon}{n(1 + |a|)^{n-1}}.$$
Hence, given $\epsilon > 0$, I choose
$$\delta = \min\left\{1,\frac{\epsilon}{n(1 + |a|)^{n-1}}\right\}.$$
For all $x$, $0 < |x - a| < \delta$ implies $0 < |x - a| < 1$ and $0 < |x - a| < \epsilon/[n(1 + |a|)^{n-1}]$, in which case
$$|x^n - a^n| < n(1 + |a|)^{n-1}|x - a| < \epsilon.$$
Since $\epsilon$ and $a$ were arbitrary, $\lim\limits_{x\to a} x^n = a^n$ for all real numbers $a$.