real exchange rate appreciation i find the topic of real exchange rate appreciation / depreciation often not very well explained. 

even the excerpt below from the following accepted answer on this website is not correct in my opinion. 


  The home currency appreciates in real terms against a foreign currency either if the home currency appreciates in nominal terms or if the home country's inflation rate is lower than that in the foreign currency.


specifically, the part in bold i believe is wrong. if the home country's inflation rate is lower than that of the foreign country, then all else equal, the home currency should depreciate in real terms as the domestic consumption basket becomes cheaper relative to the foreign consumption basket

some of the confusion abounds because of the different quoting conventions (X vs 1/X, foreign vs domestic, etc)
You need a clear understanding of what is exchange rate and how it is determined. 

Consider two countries with free flow of goods and services that is there is no transaction, transportation, tariff costs etc. [I am assuming an ideal world here].

say, 

India (Currency: Indian Rupee abbreviated as INR) and USA (Currency: Dollar abbreviated as $ ).

Exchange rate:

$ 1 = INR 50

Consider a commodity let us say pencils:

Assume that in year 2018 - one pencil costs $ 1 in USA and INR 50 in India.

Now let us suppose that India faces an inflation of 10% per annum and there is no inflation in USA.

Hence new prices of pencil in year 2019:

India - INR 55

USA - $ 1

Since, there are no transportation and transaction costs Indians will start purchasing pencil from USA since 1 US $ still equals INR 50 and Indians get pencils relatively cheaper in USA after converting their currency into dollars. This will push up Indian imports and hence Indian currency will depreciate and US dollar will appreciate because US exports have gone up.

This process will continue till 1$ = INR 55.

Now coming back to your question say home country is USA and due to relatively lower inflation rate in USA, the US dollar appreciates (NOT depreciate).