It is a part of the proof of lower semi-continuity of mapping $x \mapsto \|x\|_{bv}$, where $x\colon[0,T]\to E$, $(E,d)$ is a metric space, and the norm is defined as $\|x\|_{bv}:=\sup_{t_{i}\in\mathcal{P}}\sum d(x_{t_{i}},x_{t_{i+1}})$, $\mathcal{P}$ is a partition of $[0,T]$.
The theory states:
If $\{x^{n}\}$ is a sequence of mapping from $[0,T]$ to $E$ of bounded variation (with $\|x^{n}\|_{bv}<\infty$) and $x^{n}\to x$ pointwise on $[0,T]$, then for all $0\leq s
: $\|x\|_{bv}\leq \lim_{n\rightarrow \infty} \inf\|x^{n}\|_{bv}.$
The proof is given as:
Let $P = \{0=t_{0}
$\sum_{i=0}^{K-1}d(x_{t_{i}},x_{t_{i+1}})=\lim_{n\rightarrow \infty} \sum_{i=0}^{K-1}d(x_{t_{i}}^{n},x_{t_{i+1}}^{n})=\lim_{n\rightarrow \infty} \inf\sum_{i=0}^{K-1}d(x_{t_{i}}^{n},x_{t_{i+1}}^{n})\leq\lim_{n\rightarrow \infty} \inf\|x^{n}\|_{bv}.$
I am stuck at the first equality, how to prove
$\left|\sum_{i=0}^{K-1}d(x_{t_{i}},x_{t_{i+1}})-\sum_{i=0}^{K-1}d(x_{t_{i}}^{n},x_{t_{i+1}}^{n})\right| \rightarrow 0$ as $n\rightarrow \infty$.
P.S: if my statement was not so clear, please feel free to point out.