I just need to know if I'm correctly going about determining whether the following functions converge to their pointwise limits uniformly: \begin{align*} &(1) f_n(x) = \frac{nx}{\cos(\frac{x}{n}) + n^2x},\ x\in[0,1]\\ &(2) f_n(x) = 3x^{2n},\ x\in[0,1] \end{align*}

I'm stuck on $(1)$. I know that the limiting function $f$ on $[0,1]$ is $f(x) = 0$, but I'm having difficulty finding a function which converges to $0$ and also bounds $f_n(x)$.

For $(2)$, the limiting function is not continuous, and thus the convergence is not uniform.


For (1), note that $0<\cos\frac xn\leqslant1$ for $n\geqslant2$, so $$0\leqslant f_n(x)\leqslant\frac{nx}{1+n^2x}<1$$ for all $x\in[0,1]$. Since $$\sup_{x\in[0,1]}\left\{\frac{nx}{1+n^2x}\right\} = \frac n{1+n^2}\stackrel{n\to\infty}\longrightarrow 0, $$ it follows that $f_n$ converges uniformly to $0$.

  • 1
    $\begingroup$ Huh... that's more or less what I had, but was having trouble finding justification for my steps. Thank you so much! $\endgroup$
    – user341562
    Apr 1 '16 at 17:37

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