I have the following sequence of functions: $$f_k(x)=kx\sin\left(\frac{k^2x^2}{k+k^3x^2}\right)$$ and want to show that it converges to $x$ on [0,1] in a monotone way. I am able to show the pointwise convergence to $x$. However, I am not sure how to show that the sequence is monotone. I know that I have to show that $f_{k+1}\leq f_{k}$ or $f_{k}\leq f_{k+1}$. Can anyone help?
2026-02-24 23:19:49.1771975189
How to show a sequence is monotone
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Recall that the product of two positive monotone functions (both non-increasing or both nondecreasing) is monotone. Observe that $nx$ is monotone and $\sin$ is also monotone (both nondecreasing) and positive on $[0,1]$ and so the product must be monotone. See Monotonicity of the sum/product/max of two monotone functions.