A Eurozone nation like Greece that runs a persistent current account deficit cannot sustain rising living standards over time given that the ECB chooses to maintain rigid control of the inflation rate.
Answer: False
The answer is False.
While the answer is False the reality is that it is highly unlikely that such a nation could achieve higher living standards within the EMU.
The Euro area pverall is currently running a trade balance with the rest of the world (Source). However, within the Euro area there are sharp disparities in trade outcomes. You can get access to all the intra and extra EU trade statistics from Eurostat.
What you learn is that Germany runs a trade surplus whereas other nations such as Portugal, Greece and Spain run trade deficits.
The nations that are running trade deficits are increasing their indebtedness to the surplus nations because the exchange rate cannot adjust. This debt is in a currency that the governments of each country do not issue but which is controlled by the central banking system of the EMU.
In times of stress, the cost of accessing this debt rises and ultimately the creditor nations will stop issuing new debt to the external deficit nations. In a floating exchange rate system, the external deficits are always putting downward pressure on the exchange rate which is offset by these capital flows. If the capital flows stop, then the trade deficit will drive a depreciation and that sets in train a process of resolution because it renders the economy more competitive. Its exports become more attractive and its imports become more expensive.
In the case of an EMU nation, the only policy approach left to them to avoid persistent intra-EMU deficits is to contract fiscal policy and try to engineer an internal devaluation thereby cutting living standards.
With the ECB maintaining an overall deflationary monetary policy stance across the EMU region, this further impacts exacerbates the situation. Higher rates means the debt burden is higher overall which impacts more severely on nations that are already being forced to grow more slowly.
So typically the proposition in the question is true but not impossible.
It is possible (but unlikely) if, say, Greece could run large and dominant external deficits with non-Euro nations. Then if those nations wished to accumulate financial assets denominated in Euros then they would be willing to extend credit to say Greece. The Greek external deficits would serve to "fund" the desire by the foreigners to have Euros and they would be willing, in return, to net ship real goods and services to Greece. This scenario is highly improbable though.
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