Whats wrong in my subsequence validation?

Your loop on j (for j in range(i, len(array)):) starts at an index that corresponds to a position in sequence. Advancing through array must be independent of the index of values in sequence.

You can do this with an iterator:

array=[5, 1, 22, 25, 6, -1, 8, 10] 
sequence= [1, 6, -1, 10]

iArray = iter(array)
isSubsequence = all(s in iArray for s in sequence)

print(isSubsequence) # True

The s in iArray part advances the iterator (iArray) up to the point where s is found. If array contains all the elements of sequence in the same order, then every s in sequence will be able to advance to a matching value in iArray.