Let $(X, d)$ be a metric space, and $\gamma: [a,b]\to X$ be a curve. For any partition $P=\{a=y_0
We denote: $\|P\|=\max_i|y_i-y_{i+1}|$
Now my question is the proof of the following statement: $\lim_{\|P\|\to 0}\Sigma(P)=L(\gamma)$
The hard part for me to prove the statement is if $P$ and $Q$ are two partitions, with $\|P\|\le \|Q\|$, we only know $\Sigma(P\cup Q)\ge\max(\Sigma(P), \Sigma(Q))$ and this won't give me any contradiction when we assume there is a sequence of partitions say $P_i$ with $\|P_i\|\to 0$ and $\Sigma(P_i)\le L(\gamma)-\varepsilon_0$ for some fixed $\varepsilon_0>0$. Anybody can help?
(btw. I thought that this may be similar with the proof of the Riemann sum for integrable function, but there one has the Osilation)