Hint: look at 500.

Easy combo. Let's define $2$ killers for each point. Call a a sequence of numbers orignating from the number killer if its the smallest sequence such that the sum of the numbers in it is greater than $100000$. Consider the killer of $500$, then removing $500$ from its killer we still get a number greater than $100000$ (otherwise we are done), this must be less than $100500$ so done.

We claim some $400$ consecutive numbers satisfy this. Notice that $601+602+\dots+1000>600\cdot 400>100000$ and $1+2+\dots+400=401\cdot 200<100000$. Shifting the sum down by $1$ will decrease it by $400$ so at some point we must land in the desired range.