Test pointwise and uniform convergence of the following sequences of functions on $[0,1]$: $f_{j}=x^{j}-x^{j+1}$ and $f_{j}=x^{j}-x^{2j}$.

Question

Test pointwise and uniform convergence of the following sequences functions on $[0,1]$: $$f_{j}=x^{j}-x^{j+1}\;\;\text{ and }\;\;f_{j}=x^{j}-x^{2j}.$$

I've proved that both converge uniformly to $0$, however I'm doubting myself as my teacher wouldn't give two almost equal examples.

It's very easy to see that $\forall x \in [0,1], f_{j}$ tends to $0$ in both cases, and moreover that $\lim_{j\rightarrow \infty}\sup_{x\in [0,1]}| f_{j}-0 | =0$, so in both cases the convergence is uniform.

Answer 1

The second sequence does NOT converge uniformly to $0$ on $[0,1]$. We have $\lim_{j\to\infty}(1-1/j)^j-(1-1/j)^{2j}=e^{-1}-e^{-2}>0.$ So for all but finitely many $j\in\Bbb N$ we have $$\sup_{x\in [0,1]}x^j-x^{2j}\ge (1-1/j)^j-(1-1/j)^{2j}>\frac {1}{2}(e^{-1}-e^{-2}).$$