How inflation affects exchange rates? I have a question regarding the impact of inflation on exchange rate between trading countries.

Let's assume a hypothetical situation where the UK and the USA trade with each other. The USA import goods from the UK and the UK exports goods to the USA. 

Assume the exchange rate £1=$5 at the beginning. The UK is selling one widget to the USA for £10. Now let's imagine an inflation of 100% happened so that the UK prices this widget £20.

Now, my reasoning is that £ should weaken against dollar so that £1 = \$2.5.
So inflation weakens country's currency.
My reasoning is that this is logical that this should happen because now £1 is worth less than it was before the inflation (with the same amount of pounds you could buy twice as much before the inflation than you can now) so £1 should cost \$2.5 instead of \$5. Is my reasoning correct?
Consider two standard relations that tend to hold:

$$i_t = r_t + \pi^e_{t+1} \tag{1}$$

where $i_t$ is domestic nominal interest rate, $r_t$ is domestic real interest rate, and $\pi^e_{t+1}$ is expected domestic inflation. The expression is known as the Fisher equation.  

Also, we have the uncovered interest rate parity

$$ i_t  = i^*_t - [s^e_{t+1} - s_t] \tag{2}$$

where $i^*_t$ is "foreign" nominal interest rate, and $[s^e_{t+1} - s_t]$ is expected appreciation of domestic currency in percentage terms ($s$ is the natural logarithm of "foreign currency per unit of domestic currency" exchange rate). 
Combining the two and re-arranging we have

$$[s^e_{t+1} - s_t] = i^*_t - r_t - \pi^e_{t+1} \tag{3}$$

Assume that inflation is monetary, and keep  $i^*_t,r_t$ constant. Then expectations for domestic inflation will tend to reduce expected appreciation of domestic currency, and may even turn it into an expected depreciation, especially if expected domestic inflation is high. Shortening the length of the time periods, this tendency becomes more of "actual" inflation and "actual" exchange rate.  

The fundamental reason is of course what the OP was thinking: if we think of USD as another good (which it is since it is a store of value and can be exchanged for goods), then, if English pound loses value due to inflation and can buy less goods, why should it be still able to buy the same amount of the good "USD"?