How does one lock their own Bitcoins for X amount of blocks and prove that you locked it? Is there a method of locking Bitcoins for X amount of blocks and also being able to prove its locked? How would one do this and with what wallet? Mobile wallet possible?

Reason why is I'd imagine one can build some interesting applications for this if you can PROVE you locked the coins for X amount of time. What if there was some type of bounty reward paid if X amount of bitcoins was locked for say 1-5 years?
You could send your Bitcoin to an address that has a redeem/witness script containing OP_CHECKSEQUENCEVERIFY (for a relative timelock) or OP_CHECKLOCKTIMEVERIFY (for absolute timelock). Once the transaction has confirmed, the output cannot be spent without the timelocks being met first.

By revealing the script, you can show that those outputs have a locktime and thus cannot be spent until after that locktime passes.