Related to sequence of function

Question

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I've done the options B, C, Dand B is uniformly convergent. But what about option A, I am unable to check weather it's convergence uniform or not?? Thanks.

Answer 1

$(1-|x|)^{n} \to 0$ pointwise. But $(1-\frac 1{n^{2}})^{n}=((1-\frac 1{n^{2}})^{n^{2}})^{\frac n {n^{2}}} \to (\frac 1 e)^{0}=1$. Hence the convergence is not uniform.

Details: suppose the sequence converges uniformly. Let $0<\epsilon <1$. Since the pointwise limit is $0$ it follows that the uniform limit is also $0$. By definition, there exists $n_0$ such that $(1-|x|)^{n}<\epsilon$ for all $x \in (-1,1)$ for all $n \geq n_0$. In particular $(1-\frac 1 {n^{2}})^{n}<\epsilon$ for all $n \geq n_0$. Letting $n \to \infty$ we get $1 \leq \epsilon$ which is a contradiction.