To solve this inequality, we can first find a common denominator for all the fractions:

1/21 + 1/22 + 1/23 + ..... + 1/60 = (60*59*58*...*22*21 + 60*59*58*...*23*21 + ... + 60*59*58*...*21*22) / (21*22*23*...*60)

Now, we can simplify the numerator:

60*59*58*...*22*21 + 60*59*58*...*23*21 + ... + 60*59*58*...*21*22 = (60*59*58*...*21)(1/22 + 1/23 + ... + 1/60)

So, the inequality becomes:

(60*59*58*...*21)(1/22 + 1/23 + ... + 1/60) / (21*22*23*...*60) > 5/6

Now, we can simplify the expression:

(1/22 + 1/23 + ... + 1/60) / (21*22*23*...*60) > 5 / (6*60*59*58*...*21)

Now, we can see that the left side of the inequality is a sum of fractions, each of which is less than 1/21. Therefore, the left side of the inequality is less than (1/21)*(40) = 40/21.

So, we have:

40/21 > 5 / (6*60*59*58*...*21)

Solving for the right side of the inequality, we get:

40/21 > 5 / (6*60*59*58*...*21)
40*6*60*59*58*...*21 > 21*5
40*6*60*59*58*...*21 > 105
40*6*60*59*58*...*21 > 105

Therefore, the inequality 1/21 + 1/22 + 1/23 + ..... + 1/60 > 5/6 is true.