Convergence and topology Please what is the classical method to answer this question, does the sequence converge in the given topology ? 
1) The sequence $\big(1+(-1)^n\big)_{n\in\mathbb N}$ in $(\mathbb{R},\tau)$ such that $\tau=\{U\subset \mathbb{R},\ 0\in U\}$.
2) The sequence $\big((\frac1n,1-\frac1n,\frac{n}{n+1})\big)_{n\in\mathbb{N}^*}$ in $X=[0,1]\times[0,1]\times[0,1]$ with the usual topology in $\mathbb{R}$.
I know that: $(u_n)$ converge to $\ell$ in a topological space $E$ if $$\forall V\in \mathcal{V}_\ell,\ \exists n_0\in\mathbb{N},\ \forall n,\ n\geq n_0\Rightarrow x_n\in V
$$
 but I don't know how to apply it !
Thank you

Hint:
First you need to know what the open set looks like. 
For example, the open sets in Q1 is any subset of $\mathbb{R}$ containing $0$, for example $[0,2]$ is open. Then by definition, can you find any $x\in\mathbb{R}$ such that every open set containing $x$ containing tails of the sequence, hence your sequence is convergent? (ps, I think $\mathcal{T}$ you given is actually NOT a topology, you need to include $\emptyset$) 
Also note the topological space is not Hausdorff, so the limit may not be unique.

 for example $\forall x= 2$ is a limit.

For Q2, it's the product topology on $X=[0,1]^3$, you can use the fact that the sequence is convergent iff each component is convergent in $[0,1]$. OR, you can directly check by the definition.
