Integer expression expected

Your assignment X=$X+1 doesn't perform arithmetic. If $X is 1, it sets it to the string "1+1". Change X=$X+1 to let X=X+1 or let X++.

As for the use of -lt rather than <, that's just part of the syntax of [ (i.e., the test command). It uses = and != for string equality and inequality -eq, -ne, -lt, -le, -gt, and -ge for numbers. As @Malvolio points out, the use of < would be inconvenient, since it's the input redirection operator.

(The test / [ command that's built into the bash shell does accept < and >, but not <= or >=, for strings. But the < or > character has to be quoted to avoid interpretation as an I/O redirection operator.)

Or consider using the equivalent (( expr )) construct rather than the let command. For example, let X++ can be written as ((X++)). At least bash, ksh, and zsh support this, though sh likely doesn't. I haven't checked the respective documentation, but I presume the shells' developers would want to make them compatible.

I would use

X=`expr $X + 1`

but that's just me. And you cannot say $X < 20 because < is the input-redirect operator.

The sum X=$X+1 should be X=$(expr $X + 1 ).

You can also use < for the comparison, but you have to write (("$X" < "20")) with the double parenthesis instead of [ $X -lt 20 ].