I have some problem as following：
Let $X$ be the space of all sequences of scalars.
Define addition and scalar (= Real numbers) multiplication in the usual way. If $x_n$, $y_n\in X$, define
$\displaystyle d(x, y) = \sum_{n=1}^{\infty} 2^{-n}\frac{|x_n - y_n|}{1+|x_n - y_n|}$.
It's already known $d$ is a metric, but I have some trobule to show the continuity of addition and continuity of scalar multiplication (i.e. to show that $X$ is topological vector space (TVS)).
I'm trying to estimate by inequality (e.g. Triangle inequality), but there is no way. How can I do this gap？ Thanks.
Personally, I find that the metric obscures whats going on and that this is easier to see once we can get rid of the metric altogether and appeal to a more general topological result. Let me show how you can do this.
First notice that the space of real sequences with the metric $d$ is nothing other than the product space $\mathbb{R}^\mathbb{N}$ equipped with the product topology (where $\mathbb{R}$ is viewed as an $\mathbb{R}$-topological vector space for its usual metric).
So now it suffices to know that a product of topological vector spaces is again a topological vector space (for the obvious addition and scalar multiplication). For this suppose $X_i$ ($i \in \Lambda$) are topological vector spaces and $X = \prod_{i \in \Lambda} X_i$. Then we can identify (via the obvious homeomorphism) $X \times X \cong \prod_{i \in \Lambda} X_i \times X_i$.
If $\oplus:X \times X \to X$ is addition in $X$ and $\oplus_i: X_i \times X_i \to X_i$ is addition in $X_i$ then (under the above identification) we have $\oplus = \prod_{i \in \Lambda} \oplus_i$. Then, since $\oplus_i$ is continuous for each $i$, it suffices to show that a product of continuous maps is continuous. This is a standard topological result. Indeed, if $f_i: E_i \to F_i$ are continuous then for any basic open set $\prod U_i \subseteq F = \prod F_i$ (where $U_i = E_i$ for all but finitely many $i$), we have
$$\bigg(\prod f_i\bigg)^{-1}\bigg(\prod U_i\bigg) = \prod f_i^{-1}(U_i)$$ which is a basic open set in $\prod E_i$.
The same idea will also give continuity of scalar multiplication with a tiny bit more work. We can identify$\prod (\mathbb{R} \times X_i) \cong \bigg(\prod \mathbb{R} \bigg) \times X$ in the obvious way and then note that $\otimes: \mathbb{R} \times X \to X$ is then a restriction of the product map $\prod \otimes_i$ (up to a composition with the obvious embedding $\mathbb{R} \hookrightarrow \prod \mathbb{R}$ given by $\lambda \mapsto (\lambda, \lambda, \lambda, \dots)$).